# How do you use inverse trig functions to solve equations?

Apr 12, 2015

$\sin x = n$ if and only if $x = \arcsin n + 2 \pi k$ for some integer $k$

or $x = \left(\pi - \arcsin n\right) + 2 \pi k$ for some integer $k$

$\cos x = n$ if and only if $x = \arccos n + 2 \pi k$ for some integer $k$

or $x = \left(\pi + \arccos n\right) + 2 \pi k$ for some integer $k$

$\tan x = n$ if and only if $x = \arctan n + \pi k$ for some integer $k$

and so on.

So,: Solve $7 \sin x - 5 = 0$

$7 \sin x - 5 = 0$
$7 \sin x = 5$
$\sin x = \frac{5}{7}$

$x = \arcsin \left(\frac{5}{7}\right) + 2 \pi k$ for integer $k$
or $\left(\pi - \arcsin \left(\frac{5}{7}\right)\right) + 2 \pi k$ for integer $k$