The arctrigonometric functions

and archyperbolic functions

compute inverses of the corresponding trigonometric and hyperbolic functions.

The arctrigonometric and archyperbolic functions are calculated in radians (1 radian = 180/$\mathrm{\pi }$ degrees).

For arctrigonometric functions that output in degrees, see trigd.

For information about expanding and simplifying trigonometric expressions, see expand, factor, combine/trig, and simplify/trig.

As the trigonometric and hyperbolic functions are not invertible over the entire complex plane, or for many of them even over the real line, it is necessary to define a principal branch for each such inverse function.  This is done by restricting the forward function to a principal domain on which it is invertible, and taking that domain as the range of the inverse function.

This process necessarily results in discontinuities in the inverse functions, which can be taken to be along line segments (called branch cuts) in the real or imaginary axes.  There is choice involved with this process, and the choices can have far reaching mathematical consequences.  See invtrig/details for more information about Maple's choices for the branch cuts of these functions.

For real arguments x, y, the two-argument function arctan(y, x), computes the principal value of the argument of the complex number $x+\mathrm{I}y$, so $-\mathrm{\pi }<\arctan \left(y,x\right)\le \mathrm{\pi }$. This function is extended to complex arguments by the formula

Operator notation can also be used for the inverse trigonometric and hyperbolic functions.  For example, (sin@@(-1))(x) (which is equivalent to ${\sin }^{\left(\mathrm{-1}\right)}\left(x\right)$ in 2-D math) evaluates to $\arcsin \left(x\right)$.

$\mathrm{arcsech}\left(1\right)$

$0$

$\mathrm{arccsch}\left(1\right)$

$\ln \left(1+\sqrt{2}\right)$

$\mathrm{arccot}\left(0\right)$

$\frac{\mathrm{\pi }}{2}$

${\sin }^{\left(-1\right)}\left(x\right)$

$\arcsin \left(x\right)$

$\cos \left(\arccos \left(x\right)\right)$

$x$

$\sin \left(\arccos \left(x\right)\right)$

$\sqrt{-x^2+1}$

$\arctan \left(1\&comma;-2\right)$

$-\arctan \left(\frac{1}{2}\right)+\mathrm{\pi }$

$\mathrm{evalf}\left(\right)$

$2.677945045$

$\mathrm{arcsinh}\left(1.2+3.4\mathrm{I}\right)$

$1.960545624+1.218868917\mathrm{I}$

$\mathrm{diff}\left(\arctan \left(x\right)\&comma;x\right)$

$\frac{1}{x^2+1}$

$\mathrm{D}\left(\mathrm{arcsech}\right)$

$z\mapsto -\frac{1}{z^2\cdot \sqrt{\frac{1}{z}-1}\cdot \sqrt{\frac{1}{z}+1}}$

$\mathrm{diff}\left(\mathrm{arcsech}\left(x\right)\&comma;x\right)$

$-\frac{1}{x^2\sqrt{\frac{1}{x}-1}\sqrt{\frac{1}{x}+1}}$

$\mathrm{int}\left(\mathrm{arcsinh}\left(x\right)\&comma;x\right)$

$x\mathrm{arcsinh}\left(x\right)-\sqrt{x^2+1}$

$\mathrm{convert}\left(\mathrm{arccosh}\left(x\right)\&comma;\ln \right)$

$\ln \left(x+\sqrt{x-1}\sqrt{x+1}\right)$