Inverse Cosine (Arccosine) $y = \arccos x$

$\arccos x$ (the inverse cosine, or arccosine) is the inverse of the cosine function $\cos$. Since $\cos$ is also periodic, it is not one-to-one over its whole domain. We therefore restrict it to the interval $[0, \pi]$ on which it is decreasing and define the inverse there (the principal value) as $\arccos$.

Definition

$y = \arccos x$ is the value $y$ satisfying $\cos y = x$ with $0 \le y \le \pi$.

Domain and range

The domain is $-1 \le x \le 1$, that is $[-1, 1]$. The range is $[0, \pi]$.

Monotonicity and symmetry

The derivative

is always negative, so $\arccos$ is strictly decreasing over its whole domain. It is neither odd nor even, but it is symmetric about the point $\left(0, \dfrac{\pi}{2}\right)$: that is, $\arccos(-x) = \pi - \arccos x$.

Notable points and tangents

The graph passes through the three points $(-1, \pi)$, $\left(0, \dfrac{\pi}{2}\right)$, and $(1, 0)$. At the endpoints $x = \pm 1$ the tangent is vertical.

Relation to other functions

With the inverse sine we have

so $\arccos x = \dfrac{\pi}{2} - \arcsin x$. The graph of $\arccos$ is therefore the graph of $\arcsin$ flipped upside down and shifted up by $\dfrac{\pi}{2}$.

Specific values

For example $\arccos 1 = 0$, $\arccos \dfrac{1}{2} = \dfrac{\pi}{3}$, $\arccos 0 = \dfrac{\pi}{2}$, and $\arccos(-1) = \pi$.

Applications

When finding the angle between two vectors from their dot product, $\arccos$ is used to recover the angle $\theta$ from the value of $\cos\theta$. It appears frequently in geometry, physics, and computer graphics.