Select All Angle Measures For Which ${ \cos \theta = \frac{1}{2} }$- { -120^{\circ}$}$- { -60^{\circ}$}$- ${ 120^{\circ}\$} - ${ 600^{\circ}\$} - ${ 660^{\circ}\$}

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 concept of solving trigonometric equations involving the cosine function, specifically focusing on finding angle measures for which $\cos \theta = \frac{1}{2}$.

Understanding the Cosine Function

The cosine function is a periodic function that oscillates between -1 and 1. The cosine function is defined as:

$$\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$$

where $\theta$ is the angle between the adjacent side and the hypotenuse.

Solving Trigonometric Equations

To solve trigonometric equations involving the cosine function, we need to find the angle measures for which the equation is satisfied. In this case, we are given the equation:

$$\cos \theta = \frac{1}{2}$$

We can start by finding the reference angle for which the cosine function is equal to $\frac{1}{2}$. The reference angle is the acute angle between the terminal side of the angle and the x-axis.

Finding the Reference Angle

The reference angle for which the cosine function is equal to $\frac{1}{2}$ is $60^{\circ}$. This is because the cosine function is equal to $\frac{1}{2}$ at $60^{\circ}$.

Finding the Angle Measures

Now that we have found the reference angle, we can find the angle measures for which the cosine function is equal to $\frac{1}{2}$. We can use the following formula to find the angle measures:

$$\theta = \cos^{-1} \left( \frac{1}{2} \right) + 360^{\circ} k$$

where $k$ is an integer.

Calculating the Angle Measures

We can calculate the angle measures by plugging in different values of $k$ into the formula. We get:

$$\theta = 60^{\circ} + 360^{\circ} k$$

We can also use the fact that the cosine function is periodic with a period of $360^{\circ}$ to find the angle measures.

Finding the Angle Measures for the Given Options

Now that we have found the general formula for the angle measures, we can find the angle measures for the given options. We are given the following options:

• $-120^{\circ}$
• $-60^{\circ}$
• $120^{\circ}$
• $600^{\circ}$
• $660^{\circ}$

We can plug in these values into the formula to find the corresponding angle measures.

Calculating the Angle Measures for the Given Options

We can calculate the angle measures for the given options as follows:

• For $-120^{\circ}$, we have:

$$\theta = 60^{\circ} + 360^{\circ} (-1) = -300^{\circ}$$

• For $-60^{\circ}$, we have:

$$\theta = 60^{\circ} + 360^{\circ} (-1) = -300^{\circ}$$

• For $120^{\circ}$, we have:

$$\theta = 60^{\circ} + 360^{\circ} (1) = 420^{\circ}$$

• For $600^{\circ}$, we have:

$$\theta = 60^{\circ} + 360^{\circ} (1) = 420^{\circ}$$

• For $660^{\circ}$, we have:

$$\theta = 60^{\circ} + 360^{\circ} (1) = 420^{\circ}$$

Conclusion

In conclusion, we have found the angle measures for which $\cos \theta = \frac{1}{2}$. We used the reference angle of $60^{\circ}$ and the formula $\theta = \cos^{-1} \left( \frac{1}{2} \right) + 360^{\circ} k$ to find the angle measures. We also used the fact that the cosine function is periodic with a period of $360^{\circ}$ to find the angle measures for the given options.

Final Answer

The final answer is:

• $-300^{\circ}$
• $-300^{\circ}$
• $420^{\circ}$
• $420^{\circ}$
• $420^{\circ}$
  Frequently Asked Questions: Solving Trigonometric Equations ===========================================================

Q: What is the reference angle for which the cosine function is equal to $\frac{1}{2}$?

A: The reference angle for which the cosine function is equal to $\frac{1}{2}$ is $60^{\circ}$.

Q: How do I find the angle measures for which the cosine function is equal to $\frac{1}{2}$?

A: To find the angle measures, you can use the formula $\theta = \cos^{-1} \left( \frac{1}{2} \right) + 360^{\circ} k$, where $k$ is an integer.

Q: What is the period of the cosine function?

A: The period of the cosine function is $360^{\circ}$.

Q: How do I find the angle measures for the given options?

A: To find the angle measures for the given options, you can plug in the values into the formula $\theta = \cos^{-1} \left( \frac{1}{2} \right) + 360^{\circ} k$.

Q: What are the angle measures for the given options?

A: The angle measures for the given options are:

• $-300^{\circ}$
• $-300^{\circ}$
• $420^{\circ}$
• $420^{\circ}$
• $420^{\circ}$

Q: Why do we need to find the reference angle?

A: We need to find the reference angle because it helps us to find the angle measures for which the cosine function is equal to $\frac{1}{2}$.

Q: How do I use the reference angle to find the angle measures?

A: To use the reference angle, you can add or subtract multiples of $360^{\circ}$ to the reference angle to find the angle measures.

Q: What is the importance of finding the angle measures?

A: Finding the angle measures is important because it helps us to understand the behavior of the cosine function and how it relates to the angles in a triangle.

Q: Can I use the sine and tangent functions to solve trigonometric equations?

A: Yes, you can use the sine and tangent functions to solve trigonometric equations. However, the cosine function is often the most convenient function to use when solving trigonometric equations.

Q: How do I know which trigonometric function to use when solving an equation?

A: To determine which trigonometric function to use, you need to examine the equation and determine which function is most relevant to the problem.

Q: Can I use trigonometric identities to solve trigonometric equations?

A: Yes, you can use trigonometric identities to solve trigonometric equations. Trigonometric identities are formulas that relate the values of different trigonometric functions.

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

• $\sin^2 \theta + \cos^2 \theta = 1$
• $\tan \theta = \frac{\sin \theta}{\cos \theta}$
• $\csc \theta = \frac{1}{\sin \theta}$
• $\sec \theta = \frac{1}{\cos \theta}$
• $\cot \theta = \frac{1}{\tan \theta}$

Q: How do I use trigonometric identities to solve trigonometric equations?

A: To use trigonometric identities, you need to examine the equation and determine which identity is most relevant to the problem. You can then use the identity to simplify the equation and solve for the unknown variable.

Conclusion

In conclusion, solving trigonometric equations is an important topic in mathematics that requires a deep understanding of the trigonometric functions and their relationships. By using the reference angle, trigonometric identities, and other techniques, you can solve trigonometric equations and gain a deeper understanding of the behavior of the trigonometric functions.