Which Of The Following Is True Of The Location Of An Angle, $\theta$, Whose Tangent Value Is $-\frac{\sqrt{3}}{3}$?A. $\theta$ Has A 30-degree Reference Angle And Is Located In Quadrant II Or IV.B. $\theta$ Has A

by ADMIN 213 views

Introduction

When dealing with trigonometric functions, it's essential to understand the relationship between the values of these functions and the location of the angle on the unit circle. In this article, we will explore the location of an angle, θ\theta, whose tangent value is −33-\frac{\sqrt{3}}{3}.

Recalling the Definition of Tangent

The tangent of an angle, θ\theta, is defined as the ratio of the sine of θ\theta to the cosine of θ\theta. Mathematically, this can be expressed as:

tan⁡(θ)=sin⁡(θ)cos⁡(θ)\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}

Understanding the Given Tangent Value

We are given that the tangent value of θ\theta is −33-\frac{\sqrt{3}}{3}. This means that:

tan⁡(θ)=−33\tan(\theta) = -\frac{\sqrt{3}}{3}

Recalling the Relationship Between Tangent and Quadrants

The tangent function is negative in Quadrants III and IV, and positive in Quadrants I and II. This means that if the tangent value of an angle is negative, the angle must be located in Quadrant III or IV.

Finding the Reference Angle

To find the reference angle, we need to find the angle whose tangent value is positive and has the same absolute value as the given tangent value. In this case, the reference angle is 30 degrees, since:

tan⁡(30∘)=33\tan(30^\circ) = \frac{\sqrt{3}}{3}

Determining the Location of the Angle

Since the tangent value of θ\theta is negative, the angle must be located in Quadrant III or IV. However, we also know that the reference angle is 30 degrees, which is a special angle that has a known sine and cosine value. Using this information, we can determine the location of the angle.

Using the Sine and Cosine Values to Determine the Location

The sine and cosine values of 30 degrees are:

sin⁡(30∘)=12\sin(30^\circ) = \frac{1}{2}

cos⁡(30∘)=32\cos(30^\circ) = \frac{\sqrt{3}}{2}

Since the tangent value of θ\theta is negative, the sine value of θ\theta must be negative, and the cosine value of θ\theta must be positive. This means that the angle must be located in Quadrant IV.

Conclusion

In conclusion, the angle θ\theta whose tangent value is −33-\frac{\sqrt{3}}{3} has a 30-degree reference angle and is located in Quadrant IV.

Additional Considerations

It's worth noting that the tangent function is periodic, meaning that the tangent value of an angle repeats every 180 degrees. This means that the angle θ\theta whose tangent value is −33-\frac{\sqrt{3}}{3} is also located in Quadrant II, but with a different reference angle.

Final Answer

The final answer is:

A. θ\theta has a 30-degree reference angle and is located in Quadrant II or IV.

Explanation

The final answer is A, because the angle θ\theta whose tangent value is −33-\frac{\sqrt{3}}{3} has a 30-degree reference angle and is located in Quadrant II or IV.

Introduction

In our previous article, we explored the location of an angle, θ\theta, whose tangent value is −33-\frac{\sqrt{3}}{3}. In this article, we will answer some frequently asked questions about the location of an angle based on its tangent value.

Q: What is the relationship between the tangent value and the location of an angle?

A: The tangent value of an angle is related to the location of the angle on the unit circle. The tangent function is negative in Quadrants III and IV, and positive in Quadrants I and II. This means that if the tangent value of an angle is negative, the angle must be located in Quadrant III or IV.

Q: How do I find the reference angle of an angle based on its tangent value?

A: To find the reference angle, you need to find the angle whose tangent value is positive and has the same absolute value as the given tangent value. For example, if the tangent value of an angle is −33-\frac{\sqrt{3}}{3}, the reference angle is 30 degrees, since tan⁡(30∘)=33\tan(30^\circ) = \frac{\sqrt{3}}{3}.

Q: What is the significance of the reference angle in determining the location of an angle?

A: The reference angle is significant in determining the location of an angle because it helps us to determine the sine and cosine values of the angle. By using the sine and cosine values, we can determine the location of the angle on the unit circle.

Q: How do I determine the location of an angle based on its sine and cosine values?

A: To determine the location of an angle based on its sine and cosine values, you need to use the following rules:

  • If the sine value is positive and the cosine value is positive, the angle is located in Quadrant I.
  • If the sine value is positive and the cosine value is negative, the angle is located in Quadrant II.
  • If the sine value is negative and the cosine value is negative, the angle is located in Quadrant III.
  • If the sine value is negative and the cosine value is positive, the angle is located in Quadrant IV.

Q: What is the relationship between the tangent function and the sine and cosine functions?

A: The tangent function is defined as the ratio of the sine function to the cosine function. Mathematically, this can be expressed as:

tan⁡(θ)=sin⁡(θ)cos⁡(θ)\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}

Q: How do I use the tangent function to determine the location of an angle?

A: To use the tangent function to determine the location of an angle, you need to follow these steps:

  1. Find the tangent value of the angle.
  2. Determine the sign of the tangent value (positive or negative).
  3. Use the sign of the tangent value to determine the location of the angle on the unit circle.

Q: What are some common tangent values and their corresponding locations on the unit circle?

A: Some common tangent values and their corresponding locations on the unit circle are:

  • tan⁡(30∘)=33\tan(30^\circ) = \frac{\sqrt{3}}{3} (Quadrant I)
  • tan⁡(45∘)=1\tan(45^\circ) = 1 (Quadrant I)
  • tan⁡(60∘)=3\tan(60^\circ) = \sqrt{3} (Quadrant I)
  • tan⁡(90∘)=∞\tan(90^\circ) = \infty (Quadrant II)
  • tan⁡(120∘)=−3\tan(120^\circ) = -\sqrt{3} (Quadrant II)
  • tan⁡(135∘)=−1\tan(135^\circ) = -1 (Quadrant II)
  • tan⁡(150∘)=3\tan(150^\circ) = \sqrt{3} (Quadrant III)
  • tan⁡(180∘)=0\tan(180^\circ) = 0 (Quadrant III)
  • tan⁡(210∘)=−3\tan(210^\circ) = -\sqrt{3} (Quadrant III)
  • tan⁡(225∘)=−1\tan(225^\circ) = -1 (Quadrant III)
  • tan⁡(240∘)=3\tan(240^\circ) = \sqrt{3} (Quadrant IV)
  • tan⁡(270∘)=0\tan(270^\circ) = 0 (Quadrant IV)
  • tan⁡(300∘)=−3\tan(300^\circ) = -\sqrt{3} (Quadrant IV)
  • tan⁡(315∘)=−1\tan(315^\circ) = -1 (Quadrant IV)

Conclusion

In conclusion, the location of an angle based on its tangent value is a fundamental concept in trigonometry. By understanding the relationship between the tangent function and the sine and cosine functions, we can determine the location of an angle on the unit circle.