Chapter 6.3, Problem 6E

(a)

To determine

To calculate: The reference angle of $99°$ .

Answer to Problem 6E

The reference angle of $99°$ is ${81}^{\circ }$ .

Explanation of Solution

Given information:

The angle $99°$ .

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 $99°$ .

Denote $\theta =99°$ .

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,

  $\begin{array}{c}\overline{\theta }=180°-99°\\ =81°\end{array}$

Thus, the reference angle of $99°$ is ${81}^{\circ }$ .

(b)

To determine

To calculate: The reference angle of $-199°$ .

Answer to Problem 6E

The reference angle of $-199°$ is ${19}^{\circ }$ .

Explanation of Solution

Given information:

The angle $-199°$ .

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 $-199°$ .

Denote $\theta =-199°$ .

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, $-199°$ and $161°$ are coterminal, so, the provided angle has terminal side in Quadrant $\text{II}$ .

The reference angle is,

  $\begin{array}{c}\overline{\theta }=180°-{161}^{\circ }\\ =19°\end{array}$

Thus, the reference angle of $-199°$ is ${19}^{\circ }$ .

(c)

To determine

To calculate: The reference angle of $359°$ .

Answer to Problem 6E

The reference angle of $359°$ is $1^{\circ }$ .

Explanation of Solution

Given information:

The angle $359°$ .

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 $359°$ .

Denote $\theta =359°$ .

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,

  $\begin{array}{c}\overline{\theta }={360}^{\circ }-359°\\ =1^{\circ }\end{array}$

Thus, the reference angle of $359°$ is $1^{\circ }$ .