# Definition:Inverse Cosine/Arccosine

## 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 $z^2 - 1$.

## Terminology

There exists the popular but misleading notation $\cos^{-1} x$, which is supposed to denote the **inverse cosine function**.

However, note that as $\cos x$ is not an injection (even though by restriction of the codomain it can be considered surjective), it does not have a well-defined inverse.

The $\arccos$ function as defined here has a well-specified image which (to a certain extent) is arbitrarily chosen for convenience.

Therefore it is preferred to the notation $\cos^{-1} x$, which (as pointed out) can be confusing and misleading.

Sometimes, $\operatorname{Cos}^{-1}$ (with a capital $\text C$) is taken to mean the same as $\arccos$.

However, this can also be confusing due to the visual similarity between that and the lower case $\text c$.

Some sources hyphenate: **arc-cosine.**

## Also denoted as

The symbol used to denote the **arccosine function** is variously seen as:

- $\arccos$
- $\operatorname {acos}$

## Sources

- 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**arc cosine, arc sine, arc tangent,**etc.**${}$**