Inverse Trigonometric Functions

Inverse trigonometric functions recover the angle from a trig ratio. The three primary ones:

• $\arcsin(y) = x$ means $\sin(x) = y$, with $x \in [-\pi/2, \pi/2]$.
• $\arccos(y) = x$ means $\cos(x) = y$, with $x \in [0, \pi]$.
• $\arctan(y) = x$ means $\tan(x) = y$, with $x \in (-\pi/2, \pi/2)$.

The restricted output range is necessary because $\sin$, $\cos$, $\tan$ are not one-to-one — many angles share the same trig ratio. By restricting the codomain, we force a unique inverse.

Notation: $\sin^{-1}(x)$ is the same as $\arcsin(x)$ — but not the same as $1/\sin(x)$ (which is $\csc x$). This notational ambiguity is a common student mistake.

Inverse trig functions appear when solving triangle problems (find the angle when sides are known), in calculus (their derivatives are tidy: $\frac{d}{dx}\arctan x = \frac{1}{1+x^2}$), and in physics (computing angles from coordinates via $\arctan2$).