# Definition:Inverse Cosine/Real/Arccosine

## Definition

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 $\left[{0 \,.\,.\, \pi}\right]$.

## Caution

There exists the a 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 an 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 $\arccos$, although this can also be confusing due to the visual similarity between that and the lower case $\text{c}$.

## Also see

- Results about
**inverse cosine**can be found here.

### Other inverse trigonometrical ratios

- Definition:Arcsine
- Definition:Arctangent
- Definition:Arccotangent
- Definition:Arcsecant
- Definition:Arccosecant

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 5$: Trigonometric Functions: Principal Values for Inverse Trigonometrical Functions - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 16.5 \ (3)$