Chapter 6.3, Problem 8E

(a)

To determine

To calculate: The reference angle of $\frac{4\pi }{3}$ .

Answer to Problem 8E

The reference angle of $\frac{4\pi }{3}$ is $\frac{\pi }{3}$ .

Explanation of Solution

Given information:

The angle $\frac{4\pi }{3}$ .

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 $\frac{4\pi }{3}$ .

Denote $\theta =\frac{4\pi }{3}$ .

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,

  $\begin{array}{c}\overline{\theta }=\frac{4\pi }{3}-\pi \\ =\frac{\pi }{3}\end{array}$

Thus, the reference angle of $\frac{4\pi }{3}$ is $\frac{\pi }{3}$ .

(b)

To determine

To calculate: The reference angle of $\frac{33\pi }{4}$ .

Answer to Problem 8E

The reference angle of $\frac{33\pi }{4}$ is $\frac{\pi }{4}$ .

Explanation of Solution

Given information:

The angle $\frac{33\pi }{4}$ .

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 $\frac{33\pi }{4}$ .

Denote $\theta =\frac{33\pi }{4}$ .

  $\begin{array}{c}\theta =\frac{33\pi }{4}\\ \theta =\frac{33\pi }{4}\cdot \frac{180}{\pi }\\ \theta ={1485}^{\circ }\end{array}$

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,

  

Since, ${1485}^{\circ }$ and $45°$ are coterminal, so, the provided angle has terminal side in Quadrant $\text{I}$ .

The reference angle is,

  $\begin{array}{c}\overline{\theta }={45}^{\circ }\\ =\frac{\pi }{4}\end{array}$

Thus, the reference angle of $\frac{33\pi }{4}$ is $\frac{\pi }{4}$ .

(c)

To determine

To calculate: The reference angle of $-\frac{23\pi }{6}$ .

Answer to Problem 8E

The reference angle of $-\frac{23\pi }{6}$ is $1^{\circ }$ .

Explanation of Solution

Given information:

The angle $-\frac{23\pi }{6}$ .

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 $-\frac{23\pi }{6}$ .

Denote $\theta =-\frac{23\pi }{6}$ .

  $\begin{array}{c}\theta =-\frac{23\pi }{6}\\ \theta =-\frac{23\pi }{6}\cdot \frac{180}{\pi }\\ \theta =-{690}^{\circ }\end{array}$

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,

  

Since, $-\frac{23\pi }{6}$ and $\frac{\pi }{6}$ are coterminal, so, the provided angle has terminal side in Quadrant $\text{I}$ .

The reference angle is,

  $\overline{\theta }=\frac{\pi }{6}$

Thus, the reference angle of $-\frac{23\pi }{6}$ is $\frac{\pi }{6}$ .