User:Fropuff/Drafts/Inverse trigonometric functions

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. These formulas are sometimes used to define the inverse trigonometric functions on the whole complex plane.

$\sin ^{-1}(z)=-i\log \left(iz+{\sqrt {1-z^{2}}}\right)$		$\csc ^{-1}(z)=-i\log \left({i \over z}+{\sqrt {1-{1 \over z^{2}}}}\right)$
$\cos ^{-1}(z)={\pi \over 2}+i\log \left(iz+{\sqrt {1-z^{2}}}\right)$		$\sec ^{-1}(z)={\pi \over 2}+i\log \left({i \over z}+{\sqrt {1-{1 \over z^{2}}}}\right)$
$\tan ^{-1}(z)={i \over 2}\log \left({\frac {1-iz}{1+iz}}\right)$		$\cot ^{-1}(z)={i \over 2}\log \left({\frac {z-i}{z+i}}\right)$

Branch cuts

Since the trigonometric functions are not one-to-one, the inverse trigonometric functions are properly multi-valued functions. In order to make them single-valued on the complex plane one must make some choice of branch cuts. The conventional choices are as follows

Function	Branch points	Branch cuts
$\sin ^{-1}$	{-1, 1, ∞}	[−∞, −1] and [1, ∞]
$\cos ^{-1}$	{-1, 1, ∞}	[−∞, −1] and [1, ∞]
$\tan ^{-1}$	{-i, i}	[−i∞, −i] and [i, i∞]
$\csc ^{-1}$	{-1, 0, 1}	[−1, 0] and [0, 1]
$\sec ^{-1}$	{-1, 0, 1}	[−1, 0] and [0, 1]
$\cot ^{-1}$	{-i, i}	[−i, 0] and [0, i]

See also

• trigonometric function
• trigonometric identities
• natural logarithm