Inverse Trigonometric Functions: Domains, Ranges and Graphs

Name: Inverse Trigonometric Functions: Domains, Ranges and Graphs
Author: StemPy Lab

Why can the sine not be undone until we cut its graph?

Learn the cuts, called principal value branches, for the six inverse functions.
Read their domains, ranges and graphs, and work out values such as $\sin^{-1}(\sin 4)$ where the angle lies outside the cut.
It covers the first part of the Indian syllabus unit (NCERT, JEE Main) and the matching topics of A-Level, IB and AP.

3 lessons, about 1 h 20 min3 simulations10 problems3 practice questionsPython in every lesson

Created by StemPy Lab

Course outline (3 lessons)

Lesson 1about 25 min
Inverse sine, inverse cosine and inverse tangent
The sine, cosine and tangent give the same value for many angles, so none of them can be undone as it stands. Each is cut to a piece where it takes every value exactly once. On $\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$ the sine has the inverse $\sin^{-1}$ , on $[0, \pi]$ the cosine has the inverse $\cos^{-1}$ , and on $\left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$ the tangent has the inverse $\tan^{-1}$ . The angle that comes out is called the principal value.
1 simulation1 practice question1 Python program3 problems
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Lesson 2about 30 min
Graphs and the other three inverse functions
The graph of an inverse function is the graph of the original reflected in the line $y = x$ . So $\sin^{-1}$ rises from $\left(-1, -\dfrac{\pi}{2}\right)$ to $\left(1, \dfrac{\pi}{2}\right)$ , $\cos^{-1}$ falls from $(-1, \pi)$ to $(1, 0)$ , and $\tan^{-1}$ rises between the two asymptotes $y = \pm\dfrac{\pi}{2}$ . The other three inverses are $\cot^{-1}$ with range $(0, \pi)$ , $\sec^{-1}$ with range $[0, \pi]$ without $\dfrac{\pi}{2}$ , and $\operatorname{cosec}^{-1}$ with range $\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$ without $0$ .
1 simulation1 practice question1 Python program3 problems1 Python lab
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Lesson 3about 25 min
Composite functions: arcsin of sin x and sin of arcsin x
The inverse undoes the function, but only inside the branch. $\sin(\sin^{-1}x) = x$ for $x$ in $[-1, 1]$ , and $\tan(\tan^{-1}x) = x$ for every real $x$ . In the other order, $\sin^{-1}(\sin x) = x$ only when $x$ is in $\left[-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right]$ . For other $x$ the result is the angle in the branch with the same sine, for example $\pi - x$ when $x$ is in $\left[\dfrac{\pi}{2}, \dfrac{3\pi}{2}\right]$ . The graphs are zigzags.
1 simulation1 practice question1 Python program4 problems1 Python lab1 question set1 note
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Domains, Ranges and Graphs: Practice Paper

Board exam format15 questions · 20 min

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Inverse Trigonometric Functions: Domains, Ranges and Graphs

Course outline (3 lessons)

Inverse sine, inverse cosine and inverse tangent

Graphs and the other three inverse functions

Composite functions: arcsin of sin x and sin of arcsin x

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