# Arccosine online calculator (acos)

You can calculate the arccosine of given numbers in degrees and radians as follows.

The acos word works as a command to calculate the arccosine values and all numbers are separated by a space. In the above example, the calculator calculates the arccosines of -1, 0.5, 0.7. The results are in degrees and radians.

The arccosine function is the inverse of cosine and the range of cosine function is $[-1,\ 1]$. So the domain of arccosine is $[-1,\ 1]$. Even if you input 999 as an argument in the form, the calculator skips it.

## The range of arccos

Because the arccosine function is the inverse of cosine, the arccosine is a multivalued function. For instance, arccos(0) is $\frac{\pi}{2}$ and $\frac{3\pi}{2}$ and $\frac{5\pi}{2}$. So it's common to set the principal value to see arccos() as a single valued function.

The principal value of arccosine is $[0,\ \pi]$.

## The arccos formula

$\textnormal{arccos}( -x ) = \pi - \textnormal{arccos}( x ) \\~\\ \textnormal{arccos}( x ) = \frac{\pi}{2} - \textnormal{arcsin}( x )$

$\textnormal{cos}( \textnormal{arccos}(x) ) = x \\~\\ \textnormal{sin}( \textnormal{arccos}(x) ) = \sqrt{ 1 - x^2 } \\~\\ \textnormal{tan}( \textnormal{arccos}(x) ) = \frac{ \sqrt{ 1 - x^2 } }{ x }$

$\textnormal{arccos}\left( \dfrac{1}{x} \right) = \textnormal{arcsec}( x )$

$\textnormal{arccos}( x ) = \dfrac{1}{2} \ \textnormal{arccos}( 2x^2 - 1 ) \quad (0 \leq x \leq 1)$