Inverse Trig Functions

Level 1 - Math I (Physics) topic page in Trigonometry.

Inverse Trigonometric Functions

Inverse trigonometric functions "undo" what the trigonometric functions do, returning the angle that corresponds to a given ratio.

Arcsine

\[\theta = \arcsin(x) \iff \sin\theta = x\]

Arccosine

\[\theta = \arccos(x) \iff \cos\theta = x\]

Arctangent

\[\theta = \arctan(x) \iff \tan\theta = x\]

Each inverse function has a restricted domain to ensure it is one-to-one (each output corresponds to exactly one input):

Arcsine: domain \([-1, 1]\), range \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\)
Arccosine: domain \([-1, 1]\), range \([0, \pi]\)
Arctangent: domain \((-\infty, \infty)\), range \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)

These restricted ranges are called the principal values.

Some important properties of inverse trigonometric functions:

Arcsin-arccos

\[\arcsin(x) + \arccos(x) = \frac{\pi}{2}\]

Arctan identity

\[\arctan(x) + \arctan\left(\frac{1}{x}\right) = \frac{\pi}{2} \quad \text{for } x > 0\]

Sine of arcsin

\[\sin(\arcsin(x)) = x \quad \text{for } x \in [-1, 1]\]

Arcsin of sine

\[\arcsin(\sin\theta) = \theta \quad \text{only if } \theta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\]

The last property shows why domain restrictions are necessary - \(\arcsin(\sin\theta)\) equals \(\theta\) only within the principal range.

Inverse trigonometric functions often appear when solving for angles in right triangles:

Arctan formula

\[\theta = \arctan\left(\frac{y}{x}\right)\]

This gives the angle whose tangent is \(y/x\), useful for converting between Cartesian and polar coordinates.