Evaluate the following expressions. Your answer must be an angle \( -\pi / 2 \leq \theta \leq \pi \) in radians, written as a multiple of \( \pi \). Note that \( \pi \) is already provided in the answer so you simply have to fill in the appropriate multiple. E.g. if the answer is \( \pi / 2 \) you should enter \( 1 / 2 \). Do not use decimal answers. Write the answer as a fraction or integer. \( \sin ^{-1}(\sin ((3 \pi / 4))= \) \( \sin ^{-1}(\sin (-\pi / 3))= \) \( \cos ^{-1}(\cos (7 \pi / 6))= \) \( \cos ^{-1}(\cos (3 \pi / 4))= \) Note: You can earn partial credit on this problem.

Let's evaluate each expression step by step. 1. **\( \sin^{-1}(\sin(3\pi/4)) \)** The range of \( \sin^{-1} \) is \( [-\pi/2, \pi/2] \). Since \( 3\pi/4 \) is outside this range, we find an equivalent angle within the range: \[ \sin(3\pi/4) = \sin(\pi - 3\pi/4) = \sin(\pi/4) \] Therefore: \[ \sin^{-1}(\sin(3\pi/4)) = \pi/4 \quad \Rightarrow \quad \frac{1}{4} \] 2. **\( \sin^{-1}(\sin(-\pi/3)) \)** The angle \( -\pi/3 \) is already within the range \( [-\pi/2, \pi/2] \). Hence: \[ \sin^{-1}(\sin(-\pi/3)) = -\pi/3 \quad \Rightarrow \quad -\frac{1}{3} \] 3. **\( \cos^{-1}(\cos(7\pi/6)) \)** The range of \( \cos^{-1} \) is \( [0, \pi] \). Since \( 7\pi/6 \) is outside this range, we find an equivalent angle within the range: \[ \cos(7\pi/6) = \cos(\pi - \pi/6) = \cos(5\pi/6) \] Therefore: \[ \cos^{-1}(\cos(7\pi/6)) = 5\pi/6 \quad \Rightarrow \quad \frac{5}{6} \] 4. **\( \cos^{-1}(\cos(3\pi/4)) \)** The angle \( 3\pi/4 \) is already within the range \( [0, \pi] \). Hence: \[ \cos^{-1}(\cos(3\pi/4)) = 3\pi/4 \quad \Rightarrow \quad \frac{3}{4} \] **Final Answers:** \[ \begin{aligned} \sin^{-1}(\sin(3\pi/4)) &= \frac{1}{4} \\ \sin^{-1}(\sin(-\pi/3)) &= -\frac{1}{3} \\ \cos^{-1}(\cos(7\pi/6)) &= \frac{5}{6} \\ \cos^{-1}(\cos(3\pi/4)) &= \frac{3}{4} \end{aligned} \]