The inverse sine, cosine and tangent functions

Notation of Inverse Trig Functions

$\sin {\theta }=r\to \arcsin {r}=\sin ^{-1}{r}=\theta$ . The domain of $y=\arcsin {x}$ is $[-1,1]$ , and its range is $[{\frac {-\pi }{2}},{\frac {\pi }{2}}]$ .

$\cos {\theta }=r\to \arccos {r}=\cos ^{-1}{r}=\theta$

$\tan {\theta }=r\to \arctan {r}=\tan ^{-1}{r}=\theta$

$\csc {\theta }=r\to \operatorname {arccsc} {r}=\csc ^{-1}{r}=\theta$

$\sec {\theta }=r\to \operatorname {arcsec} {r}=\sec ^{-1}{r}=\theta$

$\cos {\theta }=r\to \arccos {r}=\cos ^{-1}{r}=\theta$

Examples:

$\sin {\frac {\pi }{4}}={\frac {\sqrt {2}}{2}}$ , so $\arcsin {\frac {\sqrt {2}}{2}}={\frac {\pi }{4}}$ . Even though $\sin {\frac {3\pi }{4}}={\frac {\sqrt {2}}{2}}$ as well, ${\frac {3\pi }{4}}$ is outside of the range for $\arcsin {x}\theta$ .