Inverse Trigonometric Functions — Part 2: Graphs and evaluating expressions: visual lesson, CBSE Class 12 Mathematics

Lesson

Inverse Trigonometric Functions, Part 2: Graphs

Three graphs, three shapes — and every principal value is a point on one of them.

In this lesson you will

Draw the graphs of the inverse sine, cosine and tangent by reflecting the restricted graphs
Read domain, range, increasing or decreasing, and asymptotes from the graphs
Evaluate expressions such as the inverse sine of the sine of an angle using the principal branch
Evaluate a trigonometric ratio of an inverse trigonometric value with a right triangle and the correct sign
Solve simple equations involving inverse trigonometric functions

Short curves for $\sin^{-1}$ and $\cos^{-1}$ (domain $[-1, 1]$); a long one for $\tan^{-1}$ (domain $\mathbb{R}$).

Concept

Draw $y = \sin^{-1} x$ , $y = \cos^{-1} x$ , $y = \tan^{-1} x$ by reflection in $y = x$
Read domain, range, increasing/decreasing and asymptotes
Evaluate $\sin^{-1}(\sin\theta)$ and similar using the principal branch
Evaluate expressions like $\cos\left(\tan^{-1}\dfrac34\right)$ with a triangle and the right sign
Solve equations such as $\tan^{-1}(3x) = -\dfrac{\pi}{3}$

Diagram

Every point $(a, b)$ of the cosine piece gives a point $(b, a)$ of $y = \cos^{-1} x$.

If $(a, b)$ is on the restricted graph of $\cos x$ , then $(b, a)$ is on $y = \cos^{-1} x$ .