Chapter 4.1, Problem 15E

a.

To determine

To find:The coterminal angles for angle $\frac{\pi }{6}$ .

Answer to Problem 15E

The coterminal angles for angle $\frac{\pi }{6}$ are $-\frac{11\pi }{6}$ and $\frac{13\pi }{6}$ .

Explanation of Solution

Given information:

The give angle is $\frac{\pi }{6}$ .

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.

  $\begin{array}{c}\frac{\pi }{6}-2\pi =\frac{\pi -12\pi }{6}\\ =-\frac{11\pi }{6}\end{array}$

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

  $\begin{array}{c}\frac{\pi }{6}+2\pi =\frac{\pi +12\pi }{6}\\ =\frac{13\pi }{6}\end{array}$

Therefore, the coterminal angles for angle $\frac{\pi }{6}$ are $-\frac{11\pi }{6}$ and $\frac{13\pi }{6}$ .

b.

To determine

To find:The coterminal angles for angle $-\frac{5\pi }{6}$ .

Answer to Problem 15E

The coterminal angles for angle $-\frac{5\pi }{6}$ are $-\frac{17\pi }{6}$ and $\frac{7\pi }{6}$ .

Explanation of Solution

Given information:

The give angle is $-\frac{5\pi }{6}$ .

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.

  $\begin{array}{c}-\frac{5\pi }{6}-2\pi =\frac{-5\pi -12\pi }{6}\\ =-\frac{17\pi }{6}\end{array}$

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

  $\begin{array}{c}-\frac{5\pi }{6}+2\pi =\frac{-5\pi +12\pi }{6}\\ =\frac{7\pi }{6}\end{array}$

Therefore, the coterminal angles for angle $-\frac{5\pi }{6}$ are $-\frac{17\pi }{6}$ and $\frac{7\pi }{6}$ .