Values of inverse functions

Statement

Compute $\arctan(1)$, $\arccos(0)$, $\arcsin(-1/2)$.

Solution

Each inverse returns the angle, within its own interval of definition, that has the given value.

• $\arctan(1)$: the angle in $\left(-\tfrac{\pi}{2},\tfrac{\pi}{2}\right)$ with tangent $1$ is $\boxed{\dfrac{\pi}{4}}$.
• $\arccos(0)$: the angle in $[0,\pi]$ with cosine $0$ is $\boxed{\dfrac{\pi}{2}}$.
• $\arcsin(-1/2)$: the angle in $\left[-\tfrac{\pi}{2},\tfrac{\pi}{2}\right]$ with sine $-\tfrac12$ is $\boxed{-\dfrac{\pi}{6}}$.

Links

Topics: Goniometry
Concepts: Arccosine · Arcsine · Arctangent
Functions: Arccosine · Arcsine · Arctangent · Inverse trigonometric functions
Skills: Using formulae
Exercise types: Evaluating expressions