Chapter 6.3, Problem 3E

(a)

To determine

To calculate: The reference angle of $150°$ .

Answer to Problem 3E

The reference angle of $150°$ is ${30}^{\circ }$ .

Explanation of Solution

Given information:

The angle $150°$ .

Formula used:

The reference angle $\overline{\theta }$ is the angle associated with the angle $\theta$ in the standard position. It is the acute angle that is formed by the terminal side of $\theta$ and the x -axis.

Calculation:

Consider the provided angle $150°$ .

Denote $\theta =150°$ .

Recall that the reference angle $\overline{\theta }$ is the angle associated with the angle $\theta$ in the standard position. It is the acute angle that is formed by the terminal side of $\theta$ and the x -axis.

From the figure provided below,

  

The provided angle has terminal side in Quadrant $\text{II}$ .

The reference angle is,

  $\begin{array}{c}\overline{\theta }=180°-150°\\ =30°\end{array}$

Thus, the reference angle of $150°$ is ${30}^{\circ }$ .

(b)

To determine

To calculate: The reference angle of $330°$ .

Answer to Problem 3E

The reference angle of $330°$ is ${30}^{\circ }$ .

Explanation of Solution

Given information:

The angle $330°$ .

Formula used:

The reference angle $\overline{\theta }$ is the angle associated with the angle $\theta$ in the standard position. It is the acute angle that is formed by the terminal side of $\theta$ and the x -axis.

Calculation:

Consider the provided angle $330°$ .

Denote $\theta =330°$ .

Recall that the reference angle $\overline{\theta }$ is the angle associated with the angle $\theta$ in the standard position. It is the acute angle that is formed by the terminal side of $\theta$ and the x -axis.

From the figure provided below,

  

The provided angle has terminal side in Quadrant $\text{IV}$ .

The reference angle is,

  $\begin{array}{c}\overline{\theta }=360°-{330}^{\circ }\\ =30°\end{array}$

Thus, the reference angle of $330°$ is ${30}^{\circ }$ .

(c)

To determine

To calculate: The reference angle of $-30°$ .

Answer to Problem 3E

The reference angle of $-30°$ is ${30}^{\circ }$ .

Explanation of Solution

Given information:

The angle $-30°$ .

Formula used:

The reference angle $\overline{\theta }$ is the angle associated with the angle $\theta$ in the standard position. It is the acute angle that is formed by the terminal side of $\theta$ and the x -axis.

Calculation:

Consider the provided angle $-30°$ .

Denote $\theta =-30°$ .

Recall that the reference angle $\overline{\theta }$ is the angle associated with the angle $\theta$ in the standard position. It is the acute angle that is formed by the terminal side of $\theta$ and the x -axis.

From the figure provided below,

  

The provided angle has terminal side in Quadrant $\text{IV}$ .

The reference angle is,

  $\begin{array}{c}\overline{\theta }=360°-\left(-30°\right)\\ =30°\end{array}$

Thus, the reference angle of $-30°$ is ${30}^{\circ }$ .