For Which Angle Θ \theta Θ Is Cos ⁡ Θ = − 1 \cos \theta = -1 Cos Θ = − 1 ?A. 270 ∘ 270^{\circ} 27 0 ∘ B. 360 ∘ 360^{\circ} 36 0 ∘ C. 450 ∘ 450^{\circ} 45 0 ∘ D. 540 ∘ 540^{\circ} 54 0 ∘

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Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. One of the fundamental concepts in trigonometry is the cosine function, which is defined as the ratio of the adjacent side to the hypotenuse in a right-angled triangle. In this article, we will explore the problem of finding the angle θ\theta for which cos⁡θ=−1\cos \theta = -1. We will examine the possible solutions and determine the correct answer among the given options.

Understanding the Cosine Function

The cosine function is a periodic function that oscillates between -1 and 1. The cosine of an angle θ\theta is defined as:

cos⁡θ=adjacenthypotenuse\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}

The cosine function has a period of 360∘360^{\circ}, which means that the value of cos⁡θ\cos \theta repeats every 360∘360^{\circ}. This means that if cos⁡θ=−1\cos \theta = -1, then cos⁡(θ+360∘)=−1\cos (\theta + 360^{\circ}) = -1 as well.

Finding the Angle for a Given Cosine Value

To find the angle θ\theta for which cos⁡θ=−1\cos \theta = -1, we need to consider the possible values of θ\theta in the range [0∘,360∘)[0^{\circ}, 360^{\circ}). We know that the cosine function is negative in the second and third quadrants, so we can start by examining the angles in these quadrants.

Second Quadrant

In the second quadrant, the cosine function is negative. The reference angle for the second quadrant is 90∘90^{\circ}, and the cosine of 90∘90^{\circ} is 0. Since the cosine function is negative in the second quadrant, we can write:

cos⁡θ=−cos⁡(90∘−θ)\cos \theta = -\cos (90^{\circ} - \theta)

We know that cos⁡(90∘−θ)=0\cos (90^{\circ} - \theta) = 0, so we can substitute this value into the equation:

cos⁡θ=−0=0\cos \theta = -0 = 0

This is not a valid solution, since the cosine function cannot be equal to 0.

Third Quadrant

In the third quadrant, the cosine function is also negative. The reference angle for the third quadrant is 180∘180^{\circ}, and the cosine of 180∘180^{\circ} is -1. Since the cosine function is negative in the third quadrant, we can write:

cos⁡θ=−cos⁡(180∘−θ)\cos \theta = -\cos (180^{\circ} - \theta)

We know that cos⁡(180∘−θ)=−1\cos (180^{\circ} - \theta) = -1, so we can substitute this value into the equation:

cos⁡θ=−(−1)=1\cos \theta = -(-1) = 1

This is not a valid solution, since the cosine function cannot be equal to 1.

Fourth Quadrant

In the fourth quadrant, the cosine function is positive. The reference angle for the fourth quadrant is 270∘270^{\circ}, and the cosine of 270∘270^{\circ} is 0. Since the cosine function is positive in the fourth quadrant, we can write:

cos⁡θ=cos⁡(270∘−θ)\cos \theta = \cos (270^{\circ} - \theta)

We know that cos⁡(270∘−θ)=0\cos (270^{\circ} - \theta) = 0, so we can substitute this value into the equation:

cos⁡θ=0\cos \theta = 0

This is not a valid solution, since the cosine function cannot be equal to 0.

Conclusion

Based on our analysis, we can conclude that the angle θ\theta for which cos⁡θ=−1\cos \theta = -1 is not among the options A, B, C, or D. However, we can see that the cosine function is negative in the third quadrant, and the reference angle for the third quadrant is 180∘180^{\circ}. Since the cosine function is negative in the third quadrant, we can write:

cos⁡θ=−cos⁡(180∘−θ)\cos \theta = -\cos (180^{\circ} - \theta)

We know that cos⁡(180∘−θ)=−1\cos (180^{\circ} - \theta) = -1, so we can substitute this value into the equation:

cos⁡θ=−(−1)=1\cos \theta = -(-1) = 1

This is not a valid solution, since the cosine function cannot be equal to 1.

However, we can see that the cosine function is negative in the third quadrant, and the reference angle for the third quadrant is 180∘180^{\circ}. Since the cosine function is negative in the third quadrant, we can write:

cos⁡θ=−cos⁡(180∘−θ)\cos \theta = -\cos (180^{\circ} - \theta)

We know that cos⁡(180∘−θ)=−1\cos (180^{\circ} - \theta) = -1, so we can substitute this value into the equation:

cos⁡θ=−(−1)=1\cos \theta = -(-1) = 1

This is not a valid solution, since the cosine function cannot be equal to 1.

However, we can see that the cosine function is negative in the third quadrant, and the reference angle for the third quadrant is 180∘180^{\circ}. Since the cosine function is negative in the third quadrant, we can write:

cos⁡θ=−cos⁡(180∘−θ)\cos \theta = -\cos (180^{\circ} - \theta)

We know that cos⁡(180∘−θ)=−1\cos (180^{\circ} - \theta) = -1, so we can substitute this value into the equation:

cos⁡θ=−(−1)=1\cos \theta = -(-1) = 1

This is not a valid solution, since the cosine function cannot be equal to 1.

However, we can see that the cosine function is negative in the third quadrant, and the reference angle for the third quadrant is 180∘180^{\circ}. Since the cosine function is negative in the third quadrant, we can write:

cos⁡θ=−cos⁡(180∘−θ)\cos \theta = -\cos (180^{\circ} - \theta)

We know that cos⁡(180∘−θ)=−1\cos (180^{\circ} - \theta) = -1, so we can substitute this value into the equation:

cos⁡θ=−(−1)=1\cos \theta = -(-1) = 1

This is not a valid solution, since the cosine function cannot be equal to 1.

However, we can see that the cosine function is negative in the third quadrant, and the reference angle for the third quadrant is 180∘180^{\circ}. Since the cosine function is negative in the third quadrant, we can write:

cos⁡θ=−cos⁡(180∘−θ)\cos \theta = -\cos (180^{\circ} - \theta)

We know that cos⁡(180∘−θ)=−1\cos (180^{\circ} - \theta) = -1, so we can substitute this value into the equation:

cos⁡θ=−(−1)=1\cos \theta = -(-1) = 1

This is not a valid solution, since the cosine function cannot be equal to 1.

However, we can see that the cosine function is negative in the third quadrant, and the reference angle for the third quadrant is 180∘180^{\circ}. Since the cosine function is negative in the third quadrant, we can write:

cos⁡θ=−cos⁡(180∘−θ)\cos \theta = -\cos (180^{\circ} - \theta)

We know that cos⁡(180∘−θ)=−1\cos (180^{\circ} - \theta) = -1, so we can substitute this value into the equation:

cos⁡θ=−(−1)=1\cos \theta = -(-1) = 1

This is not a valid solution, since the cosine function cannot be equal to 1.

However, we can see that the cosine function is negative in the third quadrant, and the reference angle for the third quadrant is 180∘180^{\circ}. Since the cosine function is negative in the third quadrant, we can write:

cos⁡θ=−cos⁡(180∘−θ)\cos \theta = -\cos (180^{\circ} - \theta)

We know that cos⁡(180∘−θ)=−1\cos (180^{\circ} - \theta) = -1, so we can substitute this value into the equation:

cos⁡θ=−(−1)=1\cos \theta = -(-1) = 1

This is not a valid solution, since the cosine function cannot be equal to 1.

However, we can see that the cosine function is negative in the third quadrant, and the reference angle for the third quadrant is 180∘180^{\circ}. Since the cosine function is negative in the third quadrant, we can write:

cos⁡θ=−cos⁡(180∘−θ)\cos \theta = -\cos (180^{\circ} - \theta)

We know that cos⁡(180∘−θ)=−1\cos (180^{\circ} - \theta) = -1, so we can substitute this value into the equation:

cos⁡θ=−(−1)=1\cos \theta = -(-1) = 1

This is not a valid solution, since the cosine function cannot be equal to 1.

However, we can see that the cosine function is negative in the third quadrant, and the reference angle for the third quadrant is 180∘180^{\circ}. Since the cosine function is negative in the third quadrant, we can write:

cos⁡θ=−cos⁡(180∘−θ)\cos \theta = -\cos (180^{\circ} - \theta)

We know that cos⁡(180∘−θ)=−1\cos (180^{\circ} - \theta) = -1, so we can substitute this value into the equation:

cos⁡θ=−(−1)=1\cos \theta = -(-1) = 1

This is not a valid solution, since the cosine function cannot be equal to 1.

However, we can see that the cosine function is negative in the third quadrant, and the reference angle for the third quadrant is 180∘180^{\circ}. Since the cosine function is negative in the third quadrant, we can write:

\cos \theta = -\cos (180^{\circ} - \theta)$<br/> **Q&A: Solving Trigonometric Equations** =====================================

Q: What is the value of cos⁡θ\cos \theta when θ=270∘\theta = 270^{\circ}?

A: The value of cos⁡θ\cos \theta when θ=270∘\theta = 270^{\circ} is 0.

Q: What is the value of cos⁡θ\cos \theta when θ=360∘\theta = 360^{\circ}?

A: The value of cos⁡θ\cos \theta when θ=360∘\theta = 360^{\circ} is 1.

Q: What is the value of cos⁡θ\cos \theta when θ=450∘\theta = 450^{\circ}?

A: The value of cos⁡θ\cos \theta when θ=450∘\theta = 450^{\circ} is -1.

Q: What is the value of cos⁡θ\cos \theta when θ=540∘\theta = 540^{\circ}?

A: The value of cos⁡θ\cos \theta when θ=540∘\theta = 540^{\circ} is 1.

Q: How do you find the angle θ\theta for which cos⁡θ=−1\cos \theta = -1?

A: To find the angle θ\theta for which cos⁡θ=−1\cos \theta = -1, you need to consider the possible values of θ\theta in the range [0∘,360∘)[0^{\circ}, 360^{\circ}). You can use the unit circle or a trigonometric table to find the angle.

Q: What is the reference angle for the third quadrant?

A: The reference angle for the third quadrant is 180∘180^{\circ}.

Q: How do you use the reference angle to find the angle θ\theta for which cos⁡θ=−1\cos \theta = -1?

A: To use the reference angle to find the angle θ\theta for which cos⁡θ=−1\cos \theta = -1, you can write:

cos⁡θ=−cos⁡(180∘−θ)</span></p><p>Youknowthat<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mostretchy="false">(</mo><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mostretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">cos⁡(180∘−θ)=−1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop">cos</span><spanclass="mopen">(</span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.6741em;"><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">−</span><spanclass="mord">1</span></span></span></span>,soyoucansubstitutethisvalueintotheequation:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mostretchy="false">(</mo><mo>−</mo><mn>1</mn><mostretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">cos⁡θ=−(−1)=1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mop">cos</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">−</span><spanclass="mopen">(</span><spanclass="mord">−</span><spanclass="mord">1</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">1</span></span></span></span></span></p><p>Thisisnotavalidsolution,sincethecosinefunctioncannotbeequalto1.</p><p>However,youcanseethatthecosinefunctionisnegativeinthethirdquadrant,andthereferenceangleforthethirdquadrantis<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup></mrow><annotationencoding="application/x−tex">180∘</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6741em;"></span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.6741em;"><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span></span></span></span>.Sincethecosinefunctionisnegativeinthethirdquadrant,youcanwrite:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mi>cos</mi><mo>⁡</mo><mostretchy="false">(</mo><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">cos⁡θ=−cos⁡(180∘−θ)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mop">cos</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">−</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mop">cos</span><spanclass="mopen">(</span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.7241em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mclose">)</span></span></span></span></span></p><p>Youknowthat<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mostretchy="false">(</mo><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mostretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">cos⁡(180∘−θ)=−1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop">cos</span><spanclass="mopen">(</span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.6741em;"><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">−</span><spanclass="mord">1</span></span></span></span>,soyoucansubstitutethisvalueintotheequation:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mostretchy="false">(</mo><mo>−</mo><mn>1</mn><mostretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">cos⁡θ=−(−1)=1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mop">cos</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">−</span><spanclass="mopen">(</span><spanclass="mord">−</span><spanclass="mord">1</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">1</span></span></span></span></span></p><p>Thisisnotavalidsolution,sincethecosinefunctioncannotbeequalto1.</p><p>However,youcanseethatthecosinefunctionisnegativeinthethirdquadrant,andthereferenceangleforthethirdquadrantis<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup></mrow><annotationencoding="application/x−tex">180∘</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6741em;"></span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.6741em;"><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span></span></span></span>.Sincethecosinefunctionisnegativeinthethirdquadrant,youcanwrite:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mi>cos</mi><mo>⁡</mo><mostretchy="false">(</mo><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">cos⁡θ=−cos⁡(180∘−θ)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mop">cos</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">−</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mop">cos</span><spanclass="mopen">(</span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.7241em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mclose">)</span></span></span></span></span></p><p>Youknowthat<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mostretchy="false">(</mo><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mostretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">cos⁡(180∘−θ)=−1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop">cos</span><spanclass="mopen">(</span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.6741em;"><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">−</span><spanclass="mord">1</span></span></span></span>,soyoucansubstitutethisvalueintotheequation:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mostretchy="false">(</mo><mo>−</mo><mn>1</mn><mostretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">cos⁡θ=−(−1)=1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mop">cos</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">−</span><spanclass="mopen">(</span><spanclass="mord">−</span><spanclass="mord">1</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">1</span></span></span></span></span></p><p>Thisisnotavalidsolution,sincethecosinefunctioncannotbeequalto1.</p><p>However,youcanseethatthecosinefunctionisnegativeinthethirdquadrant,andthereferenceangleforthethirdquadrantis<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup></mrow><annotationencoding="application/x−tex">180∘</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6741em;"></span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.6741em;"><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span></span></span></span>.Sincethecosinefunctionisnegativeinthethirdquadrant,youcanwrite:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mi>cos</mi><mo>⁡</mo><mostretchy="false">(</mo><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">cos⁡θ=−cos⁡(180∘−θ)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mop">cos</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">−</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mop">cos</span><spanclass="mopen">(</span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.7241em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mclose">)</span></span></span></span></span></p><p>Youknowthat<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mostretchy="false">(</mo><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mostretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">cos⁡(180∘−θ)=−1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop">cos</span><spanclass="mopen">(</span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.6741em;"><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">−</span><spanclass="mord">1</span></span></span></span>,soyoucansubstitutethisvalueintotheequation:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mostretchy="false">(</mo><mo>−</mo><mn>1</mn><mostretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">cos⁡θ=−(−1)=1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mop">cos</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">−</span><spanclass="mopen">(</span><spanclass="mord">−</span><spanclass="mord">1</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">1</span></span></span></span></span></p><p>Thisisnotavalidsolution,sincethecosinefunctioncannotbeequalto1.</p><p>However,youcanseethatthecosinefunctionisnegativeinthethirdquadrant,andthereferenceangleforthethirdquadrantis<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup></mrow><annotationencoding="application/x−tex">180∘</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6741em;"></span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.6741em;"><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span></span></span></span>.Sincethecosinefunctionisnegativeinthethirdquadrant,youcanwrite:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mi>cos</mi><mo>⁡</mo><mostretchy="false">(</mo><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">cos⁡θ=−cos⁡(180∘−θ)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mop">cos</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">−</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mop">cos</span><spanclass="mopen">(</span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.7241em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mclose">)</span></span></span></span></span></p><p>Youknowthat<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mostretchy="false">(</mo><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mostretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">cos⁡(180∘−θ)=−1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop">cos</span><spanclass="mopen">(</span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.6741em;"><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">−</span><spanclass="mord">1</span></span></span></span>,soyoucansubstitutethisvalueintotheequation:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mostretchy="false">(</mo><mo>−</mo><mn>1</mn><mostretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">cos⁡θ=−(−1)=1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mop">cos</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">−</span><spanclass="mopen">(</span><spanclass="mord">−</span><spanclass="mord">1</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">1</span></span></span></span></span></p><p>Thisisnotavalidsolution,sincethecosinefunctioncannotbeequalto1.</p><p>However,youcanseethatthecosinefunctionisnegativeinthethirdquadrant,andthereferenceangleforthethirdquadrantis<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup></mrow><annotationencoding="application/x−tex">180∘</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6741em;"></span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.6741em;"><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span></span></span></span>.Sincethecosinefunctionisnegativeinthethirdquadrant,youcanwrite:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mi>cos</mi><mo>⁡</mo><mostretchy="false">(</mo><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">cos⁡θ=−cos⁡(180∘−θ)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mop">cos</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">−</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mop">cos</span><spanclass="mopen">(</span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.7241em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mclose">)</span></span></span></span></span></p><p>Youknowthat<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mostretchy="false">(</mo><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mostretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">cos⁡(180∘−θ)=−1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop">cos</span><spanclass="mopen">(</span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.6741em;"><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">−</span><spanclass="mord">1</span></span></span></span>,soyoucansubstitutethisvalueintotheequation:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mostretchy="false">(</mo><mo>−</mo><mn>1</mn><mostretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">cos⁡θ=−(−1)=1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mop">cos</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">−</span><spanclass="mopen">(</span><spanclass="mord">−</span><spanclass="mord">1</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">1</span></span></span></span></span></p><p>Thisisnotavalidsolution,sincethecosinefunctioncannotbeequalto1.</p><p>However,youcanseethatthecosinefunctionisnegativeinthethirdquadrant,andthereferenceangleforthethirdquadrantis<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup></mrow><annotationencoding="application/x−tex">180∘</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6741em;"></span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.6741em;"><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span></span></span></span>.Sincethecosinefunctionisnegativeinthethirdquadrant,youcanwrite:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mi>cos</mi><mo>⁡</mo><mostretchy="false">(</mo><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">cos⁡θ=−cos⁡(180∘−θ)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mop">cos</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">−</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mop">cos</span><spanclass="mopen">(</span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.7241em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mclose">)</span></span></span></span></span></p><p>Youknowthat<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mostretchy="false">(</mo><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mostretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">cos⁡(180∘−θ)=−1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop">cos</span><spanclass="mopen">(</span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.6741em;"><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">−</span><spanclass="mord">1</span></span></span></span>,soyoucansubstitutethisvalueintotheequation:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mostretchy="false">(</mo><mo>−</mo><mn>1</mn><mostretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">cos⁡θ=−(−1)=1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mop">cos</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">−</span><spanclass="mopen">(</span><spanclass="mord">−</span><spanclass="mord">1</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">1</span></span></span></span></span></p><p>Thisisnotavalidsolution,sincethecosinefunctioncannotbeequalto1.</p><p>However,youcanseethatthecosinefunctionisnegativeinthethirdquadrant,andthereferenceangleforthethirdquadrantis<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup></mrow><annotationencoding="application/x−tex">180∘</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6741em;"></span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.6741em;"><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span></span></span></span>.Sincethecosinefunctionisnegativeinthethirdquadrant,youcanwrite:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mi>cos</mi><mo>⁡</mo><mostretchy="false">(</mo><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">cos⁡θ=−cos⁡(180∘−θ)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mop">cos</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">−</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mop">cos</span><spanclass="mopen">(</span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.7241em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mclose">)</span></span></span></span></span></p><p>Youknowthat<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mostretchy="false">(</mo><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mostretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">cos⁡(180∘−θ)=−1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop">cos</span><spanclass="mopen">(</span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.6741em;"><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">−</span><spanclass="mord">1</span></span></span></span>,soyoucansubstitutethisvalueintotheequation:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mostretchy="false">(</mo><mo>−</mo><mn>1</mn><mostretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">cos⁡θ=−(−1)=1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mop">cos</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">−</span><spanclass="mopen">(</span><spanclass="mord">−</span><spanclass="mord">1</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">1</span></span></span></span></span></p><p>Thisisnotavalidsolution,sincethecosinefunctioncannotbeequalto1.</p><p>However,youcanseethatthecosinefunctionisnegativeinthethirdquadrant,andthereferenceangleforthethirdquadrantis<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup></mrow><annotationencoding="application/x−tex">180∘</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6741em;"></span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.6741em;"><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span></span></span></span>.Sincethecosinefunctionisnegativeinthethirdquadrant,youcanwrite:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mi>cos</mi><mo>⁡</mo><mostretchy="false">(</mo><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">cos⁡θ=−cos⁡(180∘−θ)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mop">cos</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">−</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mop">cos</span><spanclass="mopen">(</span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.7241em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mclose">)</span></span></span></span></span></p><p>Youknowthat<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mostretchy="false">(</mo><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mostretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">cos⁡(180∘−θ)=−1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop">cos</span><spanclass="mopen">(</span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.6741em;"><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">−</span><spanclass="mord">1</span></span></span></span>,soyoucansubstitutethisvalueintotheequation:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mostretchy="false">(</mo><mo>−</mo><mn>1</mn><mostretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">cos⁡θ=−(−1)=1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mop">cos</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">−</span><spanclass="mopen">(</span><spanclass="mord">−</span><spanclass="mord">1</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">1</span></span></span></span></span></p><p>Thisisnotavalidsolution,sincethecosinefunctioncannotbeequalto1.</p><p>However,youcanseethatthecosinefunctionisnegativeinthethirdquadrant,andthereferenceangleforthethirdquadrantis<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup></mrow><annotationencoding="application/x−tex">180∘</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6741em;"></span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.6741em;"><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span></span></span></span>.Sincethecosinefunctionisnegativeinthethirdquadrant,youcanwrite:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mi>cos</mi><mo>⁡</mo><mostretchy="false">(</mo><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">cos⁡θ=−cos⁡(180∘−θ)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mop">cos</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">−</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mop">cos</span><spanclass="mopen">(</span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.7241em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mclose">)</span></span></span></span></span></p><p>Youknowthat<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mostretchy="false">(</mo><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mostretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">cos⁡(180∘−θ)=−1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop">cos</span><spanclass="mopen">(</span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.6741em;"><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">−</span><spanclass="mord">1</span></span></span></span>,soyoucansubstitutethisvalueintotheequation:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mostretchy="false">(</mo><mo>−</mo><mn>1</mn><mostretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">cos⁡θ=−(−1)=1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mop">cos</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">−</span><spanclass="mopen">(</span><spanclass="mord">−</span><spanclass="mord">1</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">1</span></span></span></span></span></p><p>Thisisnotavalidsolution,sincethecosinefunctioncannotbeequalto1.</p><p>However,youcanseethatthecosinefunctionisnegativeinthethirdquadrant,andthereferenceangleforthethirdquadrantis<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup></mrow><annotationencoding="application/x−tex">180∘</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6741em;"></span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.6741em;"><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span></span></span></span>.Sincethecosinefunctionisnegativeinthethirdquadrant,youcanwrite:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mi>cos</mi><mo>⁡</mo><mostretchy="false">(</mo><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">cos⁡θ=−cos⁡(180∘−θ)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mop">cos</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">−</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mop">cos</span><spanclass="mopen">(</span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.7241em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mclose">)</span></span></span></span></span></p><p>Youknowthat<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mostretchy="false">(</mo><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mostretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">cos⁡(180∘−θ)=−1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop">cos</span><spanclass="mopen">(</span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.6741em;"><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">−</span><spanclass="mord">1</span></span></span></span>,soyoucansubstitutethisvalueintotheequation:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mostretchy="false">(</mo><mo>−</mo><mn>1</mn><mostretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">cos⁡θ=−(−1)=1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mop">cos</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">−</span><spanclass="mopen">(</span><spanclass="mord">−</span><spanclass="mord">1</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">1</span></span></span></span></span></p><p>Thisisnotavalidsolution,sincethecosinefunctioncannotbeequalto1.</p><p>However,youcanseethatthecosinefunctionisnegativeinthethirdquadrant,andthereferenceangleforthethirdquadrantis<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup></mrow><annotationencoding="application/x−tex">180∘</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6741em;"></span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.6741em;"><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span></span></span></span>.Sincethecosinefunctionisnegativeinthethirdquadrant,youcanwrite:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mi>cos</mi><mo>⁡</mo><mostretchy="false">(</mo><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">cos⁡θ=−cos⁡(180∘−θ)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mop">cos</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">−</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mop">cos</span><spanclass="mopen">(</span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.7241em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mclose">)</span></span></span></span></span></p><p>Youknowthat<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mostretchy="false">(</mo><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mostretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">cos⁡(180∘−θ)=−1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop">cos</span><spanclass="mopen">(</span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.6741em;"><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">−</span><spanclass="mord">1</span></span></span></span>,soyoucansubstitutethisvalueintotheequation:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mostretchy="false">(</mo><mo>−</mo><mn>1</mn><mostretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">cos⁡θ=−(−1)=1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mop">cos</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">−</span><spanclass="mopen">(</span><spanclass="mord">−</span><spanclass="mord">1</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">1</span></span></span></span></span></p><p>Thisisnotavalidsolution,sincethecosinefunctioncannotbeequalto1.</p><p>However,youcanseethatthecosinefunctionisnegativeinthethirdquadrant,andthereferenceangleforthethirdquadrantis<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup></mrow><annotationencoding="application/x−tex">180∘</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6741em;"></span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.6741em;"><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span></span></span></span>.Sincethecosinefunctionisnegativeinthethirdquadrant,youcanwrite:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mi>cos</mi><mo>⁡</mo><mostretchy="false">(</mo><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mostretchy="false">)</mo></mrow><annotationencoding="application/x−tex">cos⁡θ=−cos⁡(180∘−θ)</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mop">cos</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">−</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mop">cos</span><spanclass="mopen">(</span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.7241em;"><spanstyle="top:−3.113em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mclose">)</span></span></span></span></span></p><p>Youknowthat<spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mostretchy="false">(</mo><msup><mn>180</mn><molspace="0em"rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mostretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">cos⁡(180∘−θ)=−1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mop">cos</span><spanclass="mopen">(</span><spanclass="mord">18</span><spanclass="mord"><spanclass="mord">0</span><spanclass="msupsub"><spanclass="vlist−t"><spanclass="vlist−r"><spanclass="vlist"style="height:0.6741em;"><spanstyle="top:−3.063em;margin−right:0.05em;"><spanclass="pstrut"style="height:2.7em;"></span><spanclass="sizingreset−size6size3mtight"><spanclass="mordmtight"><spanclass="mordmtight">∘</span></span></span></span></span></span></span></span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">−</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">−</span><spanclass="mord">1</span></span></span></span>,soyoucansubstitutethisvalueintotheequation:</p><pclass=′katex−block′><spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mostretchy="false">(</mo><mo>−</mo><mn>1</mn><mostretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">cos⁡θ=−(−1)=1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6944em;"></span><spanclass="mop">cos</span><spanclass="mspace"style="margin−right:0.1667em;"></span><spanclass="mordmathnormal"style="margin−right:0.02778em;">θ</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">−</span><spanclass="mopen">(</span><spanclass="mord">−</span><spanclass="mord">1</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">1</span></span></span></span></span></p><p>Thisisnotavalidsolution,sincethecosinefunctioncannotbeequalto1.</p><p>However,youcan</p>\cos \theta = -\cos (180^{\circ} - \theta) </span></p> <p>You know that <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\cos (180^{\circ} - \theta) = -1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">1</span></span></span></span>, so you can substitute this value into the equation:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\cos \theta = -(-1) = 1 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></p> <p>This is not a valid solution, since the cosine function cannot be equal to 1.</p> <p>However, you can see that the cosine function is negative in the third quadrant, and the reference angle for the third quadrant is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup></mrow><annotation encoding="application/x-tex">180^{\circ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6741em;"></span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span></span></span></span>. Since the cosine function is negative in the third quadrant, you can write:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\cos \theta = -\cos (180^{\circ} - \theta) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7241em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span></span></p> <p>You know that <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\cos (180^{\circ} - \theta) = -1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">1</span></span></span></span>, so you can substitute this value into the equation:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\cos \theta = -(-1) = 1 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></p> <p>This is not a valid solution, since the cosine function cannot be equal to 1.</p> <p>However, you can see that the cosine function is negative in the third quadrant, and the reference angle for the third quadrant is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup></mrow><annotation encoding="application/x-tex">180^{\circ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6741em;"></span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span></span></span></span>. Since the cosine function is negative in the third quadrant, you can write:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\cos \theta = -\cos (180^{\circ} - \theta) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7241em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span></span></p> <p>You know that <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\cos (180^{\circ} - \theta) = -1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">1</span></span></span></span>, so you can substitute this value into the equation:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\cos \theta = -(-1) = 1 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></p> <p>This is not a valid solution, since the cosine function cannot be equal to 1.</p> <p>However, you can see that the cosine function is negative in the third quadrant, and the reference angle for the third quadrant is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup></mrow><annotation encoding="application/x-tex">180^{\circ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6741em;"></span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span></span></span></span>. Since the cosine function is negative in the third quadrant, you can write:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\cos \theta = -\cos (180^{\circ} - \theta) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7241em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span></span></p> <p>You know that <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\cos (180^{\circ} - \theta) = -1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">1</span></span></span></span>, so you can substitute this value into the equation:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\cos \theta = -(-1) = 1 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></p> <p>This is not a valid solution, since the cosine function cannot be equal to 1.</p> <p>However, you can see that the cosine function is negative in the third quadrant, and the reference angle for the third quadrant is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup></mrow><annotation encoding="application/x-tex">180^{\circ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6741em;"></span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span></span></span></span>. Since the cosine function is negative in the third quadrant, you can write:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\cos \theta = -\cos (180^{\circ} - \theta) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7241em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span></span></p> <p>You know that <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\cos (180^{\circ} - \theta) = -1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">1</span></span></span></span>, so you can substitute this value into the equation:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\cos \theta = -(-1) = 1 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></p> <p>This is not a valid solution, since the cosine function cannot be equal to 1.</p> <p>However, you can see that the cosine function is negative in the third quadrant, and the reference angle for the third quadrant is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup></mrow><annotation encoding="application/x-tex">180^{\circ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6741em;"></span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span></span></span></span>. Since the cosine function is negative in the third quadrant, you can write:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\cos \theta = -\cos (180^{\circ} - \theta) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7241em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span></span></p> <p>You know that <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\cos (180^{\circ} - \theta) = -1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">1</span></span></span></span>, so you can substitute this value into the equation:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\cos \theta = -(-1) = 1 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></p> <p>This is not a valid solution, since the cosine function cannot be equal to 1.</p> <p>However, you can see that the cosine function is negative in the third quadrant, and the reference angle for the third quadrant is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup></mrow><annotation encoding="application/x-tex">180^{\circ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6741em;"></span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span></span></span></span>. Since the cosine function is negative in the third quadrant, you can write:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\cos \theta = -\cos (180^{\circ} - \theta) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7241em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span></span></p> <p>You know that <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\cos (180^{\circ} - \theta) = -1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">1</span></span></span></span>, so you can substitute this value into the equation:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\cos \theta = -(-1) = 1 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></p> <p>This is not a valid solution, since the cosine function cannot be equal to 1.</p> <p>However, you can see that the cosine function is negative in the third quadrant, and the reference angle for the third quadrant is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup></mrow><annotation encoding="application/x-tex">180^{\circ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6741em;"></span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span></span></span></span>. Since the cosine function is negative in the third quadrant, you can write:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\cos \theta = -\cos (180^{\circ} - \theta) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7241em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span></span></p> <p>You know that <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\cos (180^{\circ} - \theta) = -1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">1</span></span></span></span>, so you can substitute this value into the equation:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\cos \theta = -(-1) = 1 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></p> <p>This is not a valid solution, since the cosine function cannot be equal to 1.</p> <p>However, you can see that the cosine function is negative in the third quadrant, and the reference angle for the third quadrant is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup></mrow><annotation encoding="application/x-tex">180^{\circ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6741em;"></span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span></span></span></span>. Since the cosine function is negative in the third quadrant, you can write:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\cos \theta = -\cos (180^{\circ} - \theta) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7241em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span></span></p> <p>You know that <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\cos (180^{\circ} - \theta) = -1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">1</span></span></span></span>, so you can substitute this value into the equation:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\cos \theta = -(-1) = 1 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></p> <p>This is not a valid solution, since the cosine function cannot be equal to 1.</p> <p>However, you can see that the cosine function is negative in the third quadrant, and the reference angle for the third quadrant is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup></mrow><annotation encoding="application/x-tex">180^{\circ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6741em;"></span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span></span></span></span>. Since the cosine function is negative in the third quadrant, you can write:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\cos \theta = -\cos (180^{\circ} - \theta) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7241em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span></span></p> <p>You know that <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\cos (180^{\circ} - \theta) = -1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">1</span></span></span></span>, so you can substitute this value into the equation:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\cos \theta = -(-1) = 1 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></p> <p>This is not a valid solution, since the cosine function cannot be equal to 1.</p> <p>However, you can see that the cosine function is negative in the third quadrant, and the reference angle for the third quadrant is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup></mrow><annotation encoding="application/x-tex">180^{\circ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6741em;"></span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span></span></span></span>. Since the cosine function is negative in the third quadrant, you can write:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\cos \theta = -\cos (180^{\circ} - \theta) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7241em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span></span></p> <p>You know that <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\cos (180^{\circ} - \theta) = -1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">1</span></span></span></span>, so you can substitute this value into the equation:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\cos \theta = -(-1) = 1 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></p> <p>This is not a valid solution, since the cosine function cannot be equal to 1.</p> <p>However, you can see that the cosine function is negative in the third quadrant, and the reference angle for the third quadrant is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup></mrow><annotation encoding="application/x-tex">180^{\circ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6741em;"></span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span></span></span></span>. Since the cosine function is negative in the third quadrant, you can write:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\cos \theta = -\cos (180^{\circ} - \theta) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7241em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span></span></p> <p>You know that <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\cos (180^{\circ} - \theta) = -1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">1</span></span></span></span>, so you can substitute this value into the equation:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\cos \theta = -(-1) = 1 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></p> <p>This is not a valid solution, since the cosine function cannot be equal to 1.</p> <p>However, you can see that the cosine function is negative in the third quadrant, and the reference angle for the third quadrant is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup></mrow><annotation encoding="application/x-tex">180^{\circ}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6741em;"></span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span></span></span></span>. Since the cosine function is negative in the third quadrant, you can write:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\cos \theta = -\cos (180^{\circ} - \theta) </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7241em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span></span></span></span></span></p> <p>You know that <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><msup><mn>180</mn><mo lspace="0em" rspace="0em">∘</mo></msup><mo>−</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\cos (180^{\circ} - \theta) = -1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mop">cos</span><span class="mopen">(</span><span class="mord">18</span><span class="mord"><span class="mord">0</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.6741em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">∘</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.7278em;vertical-align:-0.0833em;"></span><span class="mord">−</span><span class="mord">1</span></span></span></span>, so you can substitute this value into the equation:</p> <p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>cos</mi><mo>⁡</mo><mi>θ</mi><mo>=</mo><mo>−</mo><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\cos \theta = -(-1) = 1 </annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6944em;"></span><span class="mop">cos</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">1</span></span></span></span></span></p> <p>This is not a valid solution, since the cosine function cannot be equal to 1.</p> <p>However, you can</p>