Inverse Trigonometric Functions — Part 1: Principal value branches: visual lesson, CBSE Class 12 Mathematics

Lesson

Inverse Trigonometric Functions, Part 1: Principal values

Trigonometry gives a ratio from an angle. Now we go backwards: from a ratio to one angle.

In this lesson you will

Explain why trigonometric functions must be restricted before they can be inverted
State the domain and principal value branch of all six inverse trigonometric functions
Find principal values for standard inputs, positive and negative
Find the domain of expressions such as the inverse sine of a linear expression

Swap the coordinates: $P\left(\dfrac{\pi}{6}, \dfrac12\right)$ becomes $Q\left(\dfrac12, \dfrac{\pi}{6}\right)$.

Concept

Explain why $\sin x$ , $\cos x$ , $\tan x$ need a restricted domain to have inverses
State the domain and principal value branch of all six inverse functions
Find principal values, including for negative inputs
Find the domain of expressions like $\sin^{-1}(3x - 2)$

Diagram

Trigonometric functions take the "No" route.

A function can be reversed only if every output comes from exactly one input (one-one) and every target value is reached (onto).