Inverse Cosine
Inverse cosine (arccos) is the function that returns the angle whose cosine equals a given value.
Formula
\arccos(x) = \theta \text{ such that } \cos(\theta) = x, \text{ for } x \in [-1, 1]
Definition
Inverse cosine (also called arccos or $\cos^{-1}$) is the reverse of cosine: given a ratio, it finds the angle that has that cosine value. Formally, $\arccos(x)$ is the inverse of cosine restricted to $[0, \pi]$, so for any $x$ in $[-1, 1]$, $\arccos(x)$ is the unique angle $\theta$ in $[0^\circ, 180^\circ]$ with $\cos(\theta) = x$, with derivative $d/dx[\arccos(x)] = -1/\sqrt{1 - x^2}$. It satisfies the fundamental identity $\arcsin(x) + \arccos(x) = \pi/2$ for all $x$ in $[-1, 1]$.
Example
If $\cos(\theta) = 0.5$, then $\theta = \arccos(0.5) = 60^\circ$, useful whenever you know the adjacent and hypotenuse sides of a right triangle and want the angle. For a triangle with sides $a = 7$, $b = 8$, $c = 5$, the law of cosines gives $\cos(C) = (7^2 + 8^2 - 5^2)/(2 \times 7 \times 8) = 88/112 \approx 0.786$, so $C = \arccos(0.786) \approx 38.2^\circ$; this generalizes to $n$ dimensions, where the angle between vectors $u$ and $v$ satisfies $\theta = \arccos((u \cdot v)/(|u||v|))$.
Key Insight
Inverse cosine is essential for the law of cosines: once you compute $\cos(C)$, you apply arccos to find the actual angle, and its range $[0^\circ, 180^\circ]$ covers every possible angle in a triangle. The identity $\arcsin(x) + \arccos(x) = \pi/2$ reflects the complementary relationship of sine and cosine, and in information geometry, arccos of the fidelity between quantum states defines a natural distance metric on the space of pure quantum states.