Inverse Trigonometric Functions — Part 2: Properties, formulae and equations: visual lesson, ISC Class 12 Mathematics

Lesson

Inverse Trigonometric Functions, Part 2

Identities, formulae and equations: always with their conditions.

In this lesson you will

Use the rules for negative arguments and the complementary identities
Convert an inverse trigonometric expression from one function to another
Apply the sum and difference formulae for inverse tangent, sine and cosine with their conditions
Use the double and triple forms of the inverse tangent and inverse sine
Solve equations involving inverse trigonometric functions and test every root

Part 2 at a glance.

Concept

Negative arguments

$\sin^{-1}$ , $\tan^{-1}$ , $\operatorname{cosec}^{-1}$ are odd; $\cos^{-1}(-x) = \pi - \cos^{-1} x$ , and likewise for $\cot^{-1}$ and $\sec^{-1}$ .

Odd for sine, tangent and cosecant; "pi minus" for cosine, cotangent and secant.
Rule	Check at $x = 0.6$
$\sin^{-1}(-x) = -\sin^{-1} x$	holds
$\tan^{-1}(-x) = -\tan^{-1} x$	holds
$\cos^{-1}(-x) = \pi - \cos^{-1} x$	holds
$\cot^{-1}(-x) = \pi - \cot^{-1} x$	holds

Concept

Complementary sums

$\sin^{-1} x + \cos^{-1} x = \dfrac{\pi}{2}$ , $\tan^{-1} x + \cot^{-1} x = \dfrac{\pi}{2}$ , $\sec^{-1} x + \operatorname{cosec}^{-1} x = \dfrac{\pi}{2}$ .

Use these to change one inverse function into another.
Identity	Valid for	Check
$\sin^{-1} x + \cos^{-1} x = \dfrac{\pi}{2}$	$\|x\| \le 1$	$x = 0.3$ : holds
$\tan^{-1} x + \cot^{-1} x = \dfrac{\pi}{2}$	all real $x$	$x = -4$ : holds
$\sec^{-1} x + \operatorname{cosec}^{-1} x = \dfrac{\pi}{2}$	$\|x\| \ge 1$	$x = 2.5$ : holds

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