Inverse Trigonometric Functions — Part 1: Principal values and graphs: visual lesson, ISC Class 12 Mathematics

Lesson

Inverse Trigonometric Functions, Part 1

Given a ratio, which angle? A calculator always answers with one particular angle.

In this lesson you will

Explain why a trigonometric function must be restricted before it can be inverted
State the domain and principal-value range of all six inverse trigonometric functions
Find principal values, including expressions such as the inverse sine of the sine of an angle
Find the domain of a function built from an inverse trigonometric function
Sketch the graphs of the inverse sine, cosine and tangent as reflections in y = x

Part 1 at a glance.

Concept

$\sin x$ is many-one on $\mathbb{R}$ , so it has no inverse. On $\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$ it is one-one and onto $[-1, 1]$ .

Many angles share a sine; the highlighted branch keeps exactly one of them.

Concept

Each inverse function returns angles only from its principal range.