Properties and Identities of Inverse Trigonometric Functions

Name: Properties and Identities of Inverse Trigonometric Functions
Author: StemPy Lab

The inverse sine and inverse cosine of the same number add up to a right angle. Prove it, and meet its family: the reflection rules, the reciprocal rules, the sum and difference of two inverse tangents with the condition on their product, and the double angle forms. Use a right triangle to find expressions such as $\sin(\cos^{-1}x)$ . It covers the Indian syllabus (NCERT, JEE Main) and the matching topics of A-Level, IB and AP.

3 lessons, about 1 h 15 min3 simulations10 problems3 practice questionsPython in every lesson

Created by StemPy Lab

Course outline (3 lessons)

Lesson 1about 25 min
Reflection and complementary relations
The inverse functions have symmetry rules. $\sin^{-1}$ , $\tan^{-1}$ and $\operatorname{cosec}^{-1}$ are odd, so $\sin^{-1}(-x) = -\sin^{-1}x$ . For the other three, $\cos^{-1}(-x) = \pi - \cos^{-1}x$ . An inverse function and its partner add up to a right angle: $\sin^{-1}x + \cos^{-1}x = \dfrac{\pi}{2}$ for $x$ in $[-1, 1]$ , and $\tan^{-1}x + \cot^{-1}x = \dfrac{\pi}{2}$ for every real $x$ . Reciprocal rules turn $\sec^{-1}$ into $\cos^{-1}$ and $\operatorname{cosec}^{-1}$ into $\sin^{-1}$ .
1 simulation1 practice question1 Python program3 problems
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Lesson 2about 25 min
Sums and differences of arctangents
The tangent of a sum is $\dfrac{x + y}{1 - xy}$ , but the inverse tangent of this fraction equals $\tan^{-1}x + \tan^{-1}y$ only when $xy < 1$ . When $xy > 1$ the sum of two positive angles is more than $\dfrac{\pi}{2}$ , outside the branch. Then the correct value is $\pi$ more than the inverse tangent of the fraction, or $\pi$ less for two negative numbers. The difference rule has $1 + xy$ and needs $xy > -1$ .
1 simulation1 practice question1 Python program4 problems1 Python lab
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Lesson 3about 25 min
Double angles and the triangle method
A right triangle turns a trigonometric function of an inverse function into an algebraic expression: $\sin(\cos^{-1}x) = \sqrt{1 - x^2}$ and $\cos(\tan^{-1}x) = \dfrac{1}{\sqrt{1 + x^2}}$ . With $x = \tan t$ the double angle formulas give $2\tan^{-1}x$ as one inverse sine for $|x| \le 1$ , one inverse cosine for $x \ge 0$ , and one inverse tangent for $|x| < 1$ . Each form is true only on its own interval.
1 simulation1 practice question1 Python program3 problems1 Python lab1 question set1 note
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Properties and Identities: Timed Test

JEE Main format10 questions · 25 min

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Properties and Identities of Inverse Trigonometric Functions

Course outline (3 lessons)

Reflection and complementary relations

Sums and differences of arctangents

Double angles and the triangle method

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