Answer
C
Work Step by Step
A. From the definition of arctan,
x is the number from $(-\displaystyle \frac{\pi}{2}, \displaystyle \frac{\pi}{2})$ for which $\tan x=$1.
$\displaystyle \frac{3\pi}{4}\not\in (-\displaystyle \frac{\pi}{2}, \displaystyle \frac{\pi}{2}) $ so
$\displaystyle \frac{3\pi}{4}$ is NOT the solution ($-\displaystyle \frac{\pi}{4}$ is).
B. From the definition of arcsin,
x is the number from $[-\displaystyle \frac{\pi}{2}, \displaystyle \frac{\pi}{2}]$ for which $\displaystyle \sin x=\frac{\sqrt{2}}{2}.$
$\displaystyle \frac{3\pi}{4}\not\in [-\displaystyle \frac{\pi}{2}, \displaystyle \frac{\pi}{2}]$ so
$\displaystyle \frac{3\pi}{4}$ is NOT the solution.
C. From the definition of arccos,
x is the number from $[0, \pi]$ for which $\displaystyle \cos x=-\frac{\sqrt{2}}{2}.$
$\displaystyle \frac{3\pi}{4}\in [0, \pi] $ and $\displaystyle \cos\frac{3\pi}{4}=-\displaystyle \frac{\sqrt{2}}{2}$ so
$\displaystyle \frac{3\pi}{4}$ is the solution .
