# Definition:Inverse Cosine/Arccosine

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## Definition

### Real Numbers

From Shape of Cosine Function, we have that $\cos x$ is continuous and strictly decreasing on the interval $\left[{0 \,.\,.\, \pi}\right]$.

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

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

Thus from Inverse of Strictly Monotone Function, $g \left({x}\right)$ admits an inverse function, which will be continuous and strictly decreasing on $\left[{-1 \,.\,.\, 1}\right]$.

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

Thus:

- The domain of $\arccos x$ is $\left[{-1 \,.\,.\, 1}\right]$
- The image of $\arccos x$ is $\left[{0 \,.\,.\, \pi}\right]$.

### Complex Plane

The principal branch of the complex inverse cosine function is defined as:

- $\map \arccos z = \dfrac 1 i \, \map \Ln {z + \sqrt {z^2 - 1} }$

where:

- $\Ln$ denotes the principal branch of the complex natural logarithm
- $\sqrt {z^2 - 1}$ denotes the principal square root of $z^2 - 1$.