User:Fropuff/Drafts/Inverse trigonometric functions

In mathematics, the inverse trigonometric functions are the inverse functions of the trigonometric functions.

Multi-valuedness

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]

The principal branch of each of these functions maps to vertical strip of width π in the complex plane. For sin⁻¹, csc⁻¹, tan⁻¹, and cot⁻¹ the strip is conventionally chosen to be between −π/2 and π/2. For cos⁻¹ and sec⁻¹ the strip is chosen to be between 0 and π. The values of the functions on the branch cuts themselves are less widely agreed upon, and may vary from source to source.

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 complex plane. In each of the following formulas, z may be any complex number.

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

Relationship to the inverse hyperbolic functions

The inverse trigonometric functions are related to the inverse hyperbolic functions through the imaginary unit i.

$i\sin ^{-1}(z)=\sinh ^{-1}(iz)\,$		$\sin ^{-1}(iz)=i\sinh ^{-1}(z)\,$
$i\cos ^{-1}(z)=\pm \cosh ^{-1}(iz)\,$		$\cos ^{-1}(iz)=\pm i\cosh ^{-1}(z)\,$
$i\tan ^{-1}(z)=\tanh ^{-1}(iz)\,$		$\tan ^{-1}(iz)=i\tanh ^{-1}(z)\,$
$i\csc ^{-1}(z)=-\,\mathrm {csch} ^{-1}(iz)\,$		$\csc ^{-1}(iz)=-i\,\mathrm {csch} ^{-1}(z)\,$
$i\sec ^{-1}(z)=\pm \,\mathrm {sech} ^{-1}(iz)\,$		$\sec ^{-1}(iz)=\pm i\,\mathrm {sech} ^{-1}(z)\,$
$i\cot ^{-1}(z)=-\coth ^{-1}(iz)\,$		$\cot ^{-1}(iz)=-i\coth ^{-1}(z)\,$

See also

• trigonometric function
• trigonometric identities
• natural logarithm