Why the inverse functions are needed

Sine, cosine and tangent are not injective over their whole domain (they are periodic), so they have no inverses in the strict sense. To define an inverse we must restrict them to an interval on which they are injective.

Definition — The inverse trigonometric functions

• Restricting $\sin$ to $\left[-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right]$ gives a bijection with $[-1,1]$. Its inverse is $\arcsin:[-1,1]\to\left[-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right]$.
• Restricting $\cos$ to $[0,\pi]$ gives a bijection with $[-1,1]$. Its inverse is $\arccos:[-1,1]\to[0,\pi]$.
• Restricting $\tan$ to $\left(-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right)$ gives a bijection with $\mathbb{R}$. Its inverse is $\arctan:\mathbb{R}\to\left(-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right)$.

Each inverse returns the angle (in the chosen interval) whose sine, cosine or tangent equals the given number.

Links

Topics: Trigonometry
Concepts: Arccosine · Arcsine · Arctangent · Inverse function · Injectivity
Functions: Arccosine · Arcsine · Arctangent · Inverse trigonometric functions