# 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 $\closedint 0 \pi$.

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:

### 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 $z^2 - 1$.