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In mathematics , the inverse trigonometric functions are the inverse functions of the trigonometric functions .
Relationship to the natural logarithm
Just as the trigonometric functions can be expressed in terms of the exponential function , the inverse trigonometric functions can be expressed in terms of the natural logarithm . The formulas are sometimes used to define the inverse trigonometric functions on the whole complex plane .
Specifically we have
sin
−
1
(
z
)
=
−
i
log
(
i
z
+
1
−
z
2
)
{\displaystyle \sin ^{-1}(z)=-i\log \left(iz+{\sqrt {1-z^{2}}}\right)}
cos
−
1
(
z
)
=
π
2
+
i
log
(
i
z
+
1
−
z
2
)
{\displaystyle \cos ^{-1}(z)={\pi \over 2}+i\log \left(iz+{\sqrt {1-z^{2}}}\right)}
tan
−
1
(
z
)
=
i
2
log
(
1
−
i
z
1
+
i
z
)
{\displaystyle \tan ^{-1}(z)={i \over 2}\log \left({\frac {1-iz}{1+iz}}\right)}
csc
−
1
(
z
)
=
−
i
log
(
i
z
+
1
−
1
z
2
)
{\displaystyle \csc ^{-1}(z)=-i\log \left({i \over z}+{\sqrt {1-{1 \over z^{2}}}}\right)}
sec
−
1
(
z
)
=
π
2
+
i
log
(
i
z
+
1
−
1
z
2
)
{\displaystyle \sec ^{-1}(z)={\pi \over 2}+i\log \left({i \over z}+{\sqrt {1-{1 \over z^{2}}}}\right)}
cot
−
1
(
z
)
=
i
2
log
(
z
−
i
z
+
i
)
{\displaystyle \cot ^{-1}(z)={i \over 2}\log \left({\frac {z-i}{z+i}}\right)}
See also
