Finding Coterminal Angles In Exercises $15$ and $16$ , determine two coterminal angles (one positive and one negative) for each angle. Give your answers in radians. $\newline$ $15$ . (a) $\frac{\pi}{6}$ $\newline$ (b) $\frac{7 \pi}{6}$ $\newline$ $16$ . (a) $\frac{2 \pi}{3}$ $\newline$ (b) $-\frac{9 \pi}{4}$

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Coterminal and reference angles

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Q. Finding Coterminal Angles In Exercises

15

and

16

, determine two coterminal angles (one positive and one negative) for each angle. Give your answers in radians.

\newline

15

. (a)

\frac{\pi}{6}

\newline

(b)

\frac{7 \pi}{6}

\newline

16

. (a)

\frac{2 \pi}{3}

\newline

(b)

-\frac{9 \pi}{4}

Add $2\pi$ for positive coterminal angle: To find a coterminal angle, we can add or subtract multiples of $2\pi$ to the given angle. This is because adding $2\pi$ is equivalent to one full rotation around the unit circle, which brings us back to the same position.
Subtract $2\pi$ for negative coterminal angle: For the angle $\frac{\pi}{6}$ , we add $2\pi$ to find a positive coterminal angle: $\frac{\pi}{6} + 2\pi = \frac{\pi}{6} + \frac{12\pi}{6} = \frac{13\pi}{6}$ .
Add $2\pi$ for positive coterminal angle: For the angle $\frac{\pi}{6}$ , we subtract $2\pi$ to find a negative coterminal angle: $\frac{\pi}{6} - 2\pi = \frac{\pi}{6} - \frac{12\pi}{6} = -\frac{11\pi}{6}$ .
Subtract $2\pi$ for negative coterminal angle: For the angle $\frac{7\pi}{6}$ , we add $2\pi$ to find a positive coterminal angle: $\frac{7\pi}{6} + 2\pi = \frac{7\pi}{6} + \frac{12\pi}{6} = \frac{19\pi}{6}$ .
Add $2\pi$ for positive coterminal angle: For the angle $\frac{7\pi}{6}$ , we subtract $2\pi$ to find a negative coterminal angle: $\frac{7\pi}{6} - 2\pi = \frac{7\pi}{6} - \frac{12\pi}{6} = -\frac{5\pi}{6}$ .
Subtract $2\pi$ for negative coterminal angle: For the angle $\frac{2\pi}{3}$ , we add $2\pi$ to find a positive coterminal angle: $\frac{2\pi}{3} + 2\pi = \frac{2\pi}{3} + \frac{6\pi}{3} = \frac{8\pi}{3}$ .
Add $2\pi$ for positive coterminal angle: For the angle $\frac{2\pi}{3}$ , we subtract $2\pi$ to find a negative coterminal angle: $\frac{2\pi}{3} - 2\pi = \frac{2\pi}{3} - \frac{6\pi}{3} = -\frac{4\pi}{3}$ .
Subtract $2\pi$ for negative coterminal angle: For the angle $-\frac{9\pi}{4}$ , we add $2\pi$ to find a positive coterminal angle: $-\frac{9\pi}{4} + 2\pi = -\frac{9\pi}{4} + \frac{8\pi}{4} = -\frac{\pi}{4}$ .
Subtract $2\pi$ for negative coterminal angle: For the angle $-\frac{9\pi}{4}$ , we add $2\pi$ to find a positive coterminal angle: $-\frac{9\pi}{4} + 2\pi = -\frac{9\pi}{4} + \frac{8\pi}{4} = -\frac{\pi}{4}$ .For the angle $-\frac{9\pi}{4}$ , we subtract $2\pi$ to find a negative coterminal angle: $-\frac{9\pi}{4} - 2\pi = -\frac{9\pi}{4} - \frac{8\pi}{4} = -\frac{17\pi}{4}$ .