Chapter 5 - Section 5.5 - Inverse Trigonometric Functions and Their Graphs - 5.5 Exercises - Page 445: 8

a) $0$ b) $-\frac{\pi}{3}$ c) $-\frac{\pi}{6}$

a) First, we look at the interval of the inverse of tangent which is $-\frac{\pi}{2}\lt x\lt\frac{\pi}{2}$. Then, we must find what angle will give a slope of $0$ in that interval. The answer is $0$ b) First, we look at the interval of the inverse of tangent which is $-\frac{\pi}{2}\lt x\lt\frac{\pi}{2}$. Then, we must find what angle will give a slope of $-\sqrt3$ in that interval. The answer is $-\frac{\pi}{3}$ c) First, we look at the interval of the inverse of tangent which is $-\frac{\pi}{2}\lt x\lt\frac{\pi}{2}$. Then, we must find what angle will give a slope of $-\frac{\sqrt3}{3}$ in that interval. The answer is $-\frac{\pi}{6}$