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

$\arccos x$, the inverse cosine or arccosine, is the inverse of the cosine function $\cos$¹. Since $\cos$ is periodic too, it is not one-to-one over its whole domain, so it is restricted to the interval $[0, \pi]$, on which it decreases, and the inverse of that restriction, the principal value, is $\arccos$.

Definition

$y = \arccos x$ is the value $y$ satisfying both of the following.

• $\cos y = x$
• $0 \leq y \leq \pi$

Domain and range

The domain is the interval $[-1, 1]$ and the range is the interval $[0, \pi]$.

Monotonicity

The derivative is as follows.

It is always negative, so the function decreases monotonically over its whole domain. It differs from the derivative of the inverse sine only in sign, which reflects the fact that $\sin$ and $\cos$ run in opposite directions on their chosen intervals.

Symmetry

It is neither odd nor even, but it has point symmetry about $\left( 0, \dfrac{\pi}{2} \right)$.

Notable values

Tangents at the endpoints

At $x = \pm 1$ the denominator of the derivative tends to $0$, so the tangent becomes vertical. The graph drops away from $(-1, \pi)$ and comes to rest at $(1, 0)$.

Relation to the inverse sine

So $\arccos x = \dfrac{\pi}{2} - \arcsin x$, and the graph of $\arccos$ is that of $\arcsin$ turned upside down and lifted by $\dfrac{\pi}{2}$.

Applications

When the angle between two vectors is found from their dot product, $\arccos$ is what turns the value of $\cos\theta$ back into the angle $\theta$.

It appears constantly in geometry, physics and computer graphics.

1. Inverse trigonometric functions, Wikipedia