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 $[{\frac {-\pi }{2}},{\frac {\pi }{2}}]$ .

$\cos {\theta }=r\to \arccos {r}=\cos ^{-1}{r}=\theta$ . Domain: $[-1,1]$ ; range: $[0,\pi ]$ .

$\tan {\theta }=r\to \arctan {r}=\tan ^{-1}{r}=\theta$ . Domain: $(-\infty ,\infty )$ ; range: $({\frac {-\pi }{2}},{\frac {\pi }{2}})$ .

$\csc {\theta }=r\to \operatorname {arccsc} {r}=\csc ^{-1}{r}=\theta$ . Domain: $(-\infty ,-1],[1,\infty ]$ ; range: $[{\frac {-\pi }{2}},{\frac {\pi }{2}}]$ .

$\sec {\theta }=r\to \operatorname {arcsec} {r}=\sec ^{-1}{r}=\theta$ . Domain: $(-\infty ,-1],[1,\infty ]$ ; range: $[0,\pi ]$ .

$\cot {\theta }=r\to \operatorname {arccot} {r}=\cot ^{-1}{r}=\theta$ . Domain: $(-\infty ,\infty )$ ; range: $(0,\pi )$ .

Example: $\sin {\frac {\pi }{4}}={\frac {\sqrt {2}}{2}}$ , 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 $\arcsin {x}$ .

Notes: $\csc {x}=1/\sin {x}$ , and $\operatorname {arccsc} {x}=\arcsin {(1/x)}$ . $\operatorname {arccsc} {x}$ does NOT equal $1/\arcsin {x}$ . Similarly, $\operatorname {arcsec} {x}=\arccos {(1/x)}$ , and NOT $1/\arccos {x}$ .