Chapter 4.1, Problem 16E

a.

To determine

To find:The coterminal angles for angle $\frac{2\pi }{3}$ .

Answer to Problem 16E

The coterminal angles for angle $\frac{2\pi }{3}$ are $-\frac{4\pi }{3}$ and $\frac{8\pi }{3}$ .

Explanation of Solution

Given information:

The give angle is $\frac{2\pi }{3}$ .

Calculation:

Coterminal angles are defined as the angle that share side and same terminal. The coterminal angle has a positive angle and a negative angle. To determine the coterminal angles, add and subtract $2\pi$ from the given angle.

Subtract $2\pi$ from the given angle to calculate the positive angle.

  $\begin{array}{c}\frac{2\pi }{3}-2\pi =\frac{2\pi -6\pi }{3}\\ =-\frac{4\pi }{3}\end{array}$

Add $2\pi$ to the given angle to calculate the negative angle.

  $\begin{array}{c}\frac{2\pi }{3}+2\pi =\frac{2\pi +6\pi }{3}\\ =\frac{8\pi }{3}\end{array}$

Therefore, the coterminal angles for angle $\frac{2\pi }{3}$ are $-\frac{4\pi }{3}$ and $\frac{8\pi }{3}$ .

b.

To determine

To find:The coterminal angles for angle $-\frac{9\pi }{4}$ .

Answer to Problem 16E

The coterminal angles for angle $-\frac{9\pi }{4}$ are $-\frac{17\pi }{4}$ and $-\frac{\pi }{4}$ .

Explanation of Solution

Given information:

The give angle is $-\frac{9\pi }{4}$ .

Calculation:

Coterminal angles are defined as the angle that share side and same terminal. The coterminal angle has a positive angle and a negative angle. To determine the coterminal angles, add and subtract $2\pi$ from the given angle.

Subtract $2\pi$ from the given angle to calculate the positive angle.

  $\begin{array}{c}-\frac{9\pi }{4}-2\pi =\frac{-9\pi -8\pi }{4}\\ =-\frac{17\pi }{4}\end{array}$

Add $2\pi$ to the given angle to calculate the negative angle.

  $\begin{array}{c}-\frac{9\pi }{4}+2\pi =\frac{-9\pi +8\pi }{4}\\ =-\frac{\pi }{4}\end{array}$

Therefore, the coterminal angles for angle $-\frac{9\pi }{4}$ are $-\frac{17\pi }{4}$ and $-\frac{\pi }{4}$ .