# Definition:Inverse Cosine

## Definition

### Real Numbers

Let $x \in \R$ be a real number such that $-1 \le x \le 1$.

The inverse cosine of $x$ is the multifunction defined as:

$\cos^{-1} \left({x}\right) := \left\{{y \in \R: \cos \left({y}\right) = x}\right\}$

where $\cos \left({y}\right)$ is the cosine of $y$.

### Complex Plane

Let $z \in \C$ be a complex number.

The inverse cosine of $z$ is the multifunction defined as:

$\cos^{-1} \left({z}\right) := \left\{{w \in \C: \cos \left({w}\right) = z}\right\}$

where $\cos \left({w}\right)$ is the cosine of $w$.

## Arccosine

Arccosine Function

From Shape of Cosine Function, we have that $\cos x$ is continuous and strictly decreasing on the interval $\closedint 0 \pi$.

From Cosine of Multiple of Pi, $\cos \pi = -1$ and $\cos 0 = 1$.

Therefore, let $g: \closedint 0 \pi \to \closedint {-1} 1$ be the restriction of $\cos x$ to $\closedint 0 \pi$.

Thus from Inverse of Strictly Monotone Function, $\map g x$ admits an inverse function, which will be continuous and strictly decreasing on $\closedint {-1} 1$.

This function is called arccosine of $x$ and is written $\arccos x$.

Thus:

The domain of $\arccos x$ is $\closedint {-1} 1$
The image of $\arccos x$ is $\closedint 0 \pi$.