Inverse Cosine
TrigonometryInverse 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.
Example
If $\cos(\theta) = 0.5$, then $\theta = \arccos(0.5) = 60^\circ$.
Key Insight
Use inverse cosine when you know the adjacent and hypotenuse sides of a right triangle and want to find the angle.
Definition
$\arccos(x)$ is the inverse of cosine restricted to $[0, \pi]$. For any $x$ in $[-1, 1]$, $\arccos(x)$ is the unique angle $\theta$ in $[0^\circ, 180^\circ]$ such that $\cos(\theta) = x$.
Example
A triangle has sides $a = 7$, $b = 8$, $c = 5$. Using the law of cosines: $\cos(C) = (7^2 + 8^2 - 5^2)/(2 \times 7 \times 8) = 88/112 \approx 0.786$, so $C = \arccos(0.786) \approx 38.2^\circ$.
Key Insight
Inverse cosine is essential for the law of cosines: once you compute $\cos(C)$, you apply arccos to find the actual angle. The range $[0^\circ, 180^\circ]$ covers all possible angles in a triangle.
Definition
$\arccos: [-1, 1] \to [0, \pi]$ with derivative $d/dx[\arccos(x)] = -1/\sqrt{1 - x^2}$. Note $\arcsin(x) + \arccos(x) = \pi/2$ for all $x$ in $[-1, 1]$, a fundamental identity. Power series: $\arccos(x) = \pi/2 - \arcsin(x)$ for $|x| \le 1$.
Example
The angle between two vectors $u$ and $v$ in $\mathbb{R}^n$ satisfies $\cos(\theta) = (u \cdot v)/(|u||v|)$, so $\theta = \arccos((u \cdot v)/(|u||v|))$. This generalizes angle measurement to $n$-dimensional inner product spaces.
Key Insight
The identity $\arcsin(x) + \arccos(x) = \pi/2$ reflects the complementary relationship of sine and cosine. In information geometry, arccos of the fidelity between quantum states defines a natural distance metric on the space of pure quantum states.