sin θ = r → arcsin r = sin − 1 r = θ {\displaystyle \sin {\theta }=r\to \arcsin {r}=\sin ^{-1}{r}=\theta } . The domain of y = arcsin x {\displaystyle y=\arcsin {x}} is [ − 1 , 1 ] {\displaystyle [-1,1]} , and its range is [ − π 2 , π 2 ] {\displaystyle [{\frac {-\pi }{2}},{\frac {\pi }{2}}]} .
cos θ = r → arccos r = cos − 1 r = θ {\displaystyle \cos {\theta }=r\to \arccos {r}=\cos ^{-1}{r}=\theta } . Domain: [ − 1 , 1 ] {\displaystyle [-1,1]} ; range: [ 0 , π ] {\displaystyle [0,\pi ]} .
tan θ = r → arctan r = tan − 1 r = θ {\displaystyle \tan {\theta }=r\to \arctan {r}=\tan ^{-1}{r}=\theta } . Domain: ( − ∞ , ∞ ) {\displaystyle (-\infty ,\infty )} ; range: ( − π 2 , π 2 ) {\displaystyle ({\frac {-\pi }{2}},{\frac {\pi }{2}})} .
csc θ = r → arccsc r = csc − 1 r = θ {\displaystyle \csc {\theta }=r\to \operatorname {arccsc} {r}=\csc ^{-1}{r}=\theta } . ( − ∞ , − 1 ] , [ 1 , ∞ ] {\displaystyle (-\infty ,-1],[1,\infty ]} ; range: [ − π 2 , π 2 ] {\displaystyle [{\frac {-\pi }{2}},{\frac {\pi }{2}}]} .
sec θ = r → arcsec r = sec − 1 r = θ {\displaystyle \sec {\theta }=r\to \operatorname {arcsec} {r}=\sec ^{-1}{r}=\theta } . Domain: ( − ∞ , − 1 ] , [ 1 , ∞ ] {\displaystyle (-\infty ,-1],[1,\infty ]} ; range: [ 0 , π ] {\displaystyle [0,\pi ]} .
cot θ = r → arccot r = cot − 1 r = θ {\displaystyle \cot {\theta }=r\to \operatorname {arccot} {r}=\cot ^{-1}{r}=\theta } . Domain: ( − ∞ , ∞ ) {\displaystyle (-\infty ,\infty )} ; range: ( 0 , π ) {\displaystyle (0,\pi )} .
Example: sin π 4 = 2 2 {\displaystyle \sin {\frac {\pi }{4}}={\frac {\sqrt {2}}{2}}} , so arcsin 2 2 = π 4 {\displaystyle \arcsin {\frac {\sqrt {2}}{2}}={\frac {\pi }{4}}} . Even though sin 3 π 4 = 2 2 {\displaystyle \sin {\frac {3\pi }{4}}={\frac {\sqrt {2}}{2}}} as well, 3 π 4 {\displaystyle {\frac {3\pi }{4}}} is outside of the range for arcsin x {\displaystyle \arcsin {x}} .
. The domain of 
is ![{\displaystyle [-1,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51e3b7f14a6f70e614728c583409a0b9a8b9de01)
, and its range is ![{\displaystyle [{\frac {-\pi }{2}},{\frac {\pi }{2}}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9dc7f3c869691e842d0daf385da660f75dbb66c3)
.
. Domain: ; range: ![{\displaystyle [0,\pi ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e2a912eda6ef1afe46a81b518fe9da64a832751)
.
. Domain: 
; range: 
.
. ![{\displaystyle (-\infty ,-1],[1,\infty ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8181a765a6ee5ce360756b06b65a5a51382c574)
; range: .
. Domain: ; range: .
. Domain: ; range: 
.
, so 
. Even though 
as well, 
is outside of the range for 
.
