Question 1. (total 4pts, 2pts each) For each of the angle given below, find an angle between $0^{\circ}$ and $360^{\circ}$ or between 0 and $2 \pi$ that is coterminal with it.
(1.) $-475^{\circ}$
(2.) $\frac{32 \pi}{7}$
Answer
\(\boxed{\frac{4 \pi}{7}}\) is a coterminal angle with \(\frac{32 \pi}{7}\) that lies between 0 and \(2 \pi\)
Step 1 ：\(-475^\circ + 360^\circ = -115^\circ\)
Step 2 ：\(-115^\circ + 360^\circ = 245^\circ\)
Step 3 ：\(\boxed{245^\circ}\) is a coterminal angle with \(-475^\circ\) that lies between \(0^\circ\) and \(360^\circ\)
Step 4 ：\(\frac{32 \pi}{7} - 2 \pi = \frac{32 \pi}{7} - \frac{14 \pi}{7} = \frac{18 \pi}{7}\)
Step 5 ：\(\frac{18 \pi}{7} - 2 \pi = \frac{18 \pi}{7} - \frac{14 \pi}{7} = \frac{4 \pi}{7}\)
Step 6 ：\(\boxed{\frac{4 \pi}{7}}\) is a coterminal angle with \(\frac{32 \pi}{7}\) that lies between 0 and \(2 \pi\)
