Inverse Trigonometric Functions

Chapter Overview

A trigonometric function is generally periodic and hence many-to-one over its natural domain. Therefore, its inverse does not exist globally. To define inverse trigonometric functions, we restrict the domains of the standard trigonometric functions to specific intervals where they become one-to-one and onto (bijective).

The inverse function is represented as $y = \sin^{-1}x$ (or $\arcsin x$), which implies $\sin y = x$. The output $y$ represents an angle in radians, while the input $x$ is a **real number** representing the trigonometric ratio value.

Topic 1: Definitions, Domains & Principal Value Branches

The table below lists the domains and the Principal Value Branches (PVB) (or ranges) for all six inverse trigonometric functions. Solving algebraic problems and checking interval constraints relies heavily on these bounds.

Inverse Function ($y$)	Domain ($x$)	Range / Principal Value Branch ($y$)
$y = \sin^{-1}x$	$[-1, 1]$	$\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$
$y = \cos^{-1}x$	$[-1, 1]$	$[0, \pi]$
$y = \tan^{-1}x$	$\mathbb{R}$ or $(-\infty, \infty)$	$\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$
$y = \cot^{-1}x$	$\mathbb{R}$ or $(-\infty, \infty)$	$(0, \pi)$
$y = \sec^{-1}x$	$(-\infty, -1] \cup [1, \infty)$ or $\mathbb{R} \setminus (-1, 1)$	$[0, \pi] \setminus \left\{\frac{\pi}{2}\right\}$
$y = \text{cosec}^{-1}x$	$(-\infty, -1] \cup [1, \infty)$ or $\mathbb{R} \setminus (-1, 1)$	$\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \setminus \{0\}$

The Right-Angled Triangle Method (ITF Conversion)

To convert one inverse trigonometric function into another for simplified calculation or matching, use a right-angled triangle and the Pythagoras theorem ($H^2 = P^2 + B^2$):

For any $x > 0$, let $\theta = \sin^{-1}x \implies \sin\theta = \dfrac{x}{1} = \dfrac{\text{Perpendicular (P)}}{\text{Hypotenuse (H)}}$.

Perpendicular (P): $x$, Hypotenuse (H): $1$
Base (B): $\sqrt{H^2 - P^2} = \sqrt{1 - x^2}$

From this right-angled triangle, we can directly express $\theta$ in all other inverse trigonometric forms:

$\cos^{-1}\sqrt{1-x^2}$
$\tan^{-1}\left(\dfrac{x}{\sqrt{1-x^2}}\right)$
$\cot^{-1}\left(\dfrac{\sqrt{1-x^2}}{x}\right)$

$\sec^{-1}\left(\dfrac{1}{\sqrt{1-x^2}}\right)$
$\text{cosec}^{-1}\left(\dfrac{1}{x}\right)$

⚠️ Important: If $x < 0$, first use even/odd or negative properties (e.g. $\cos^{-1}(-x) = \pi - \cos^{-1}x$) to convert it to a positive value before applying the triangle method.

Topic 2: Graphs of Inverse Trigonometric Functions

These precise, high-fidelity coordinate vector graphs display domains, ranges, key intersection coordinates, and asymptotes (in dashed red) for all six functions.

$y = \sin^{-1}x$

Domain: $[-1, 1]$ | Range: $[-\frac{\pi}{2}, \frac{\pi}{2}]$

$y = \cos^{-1}x$

Domain: $[-1, 1]$ | Range: $[0, \pi]$

$y = \tan^{-1}x$

Domain: $\mathbb{R}$ | Range: $(-\frac{\pi}{2}, \frac{\pi}{2})$

$y = \cot^{-1}x$

Domain: $\mathbb{R}$ | Range: $(0, \pi)$

$y = \sec^{-1}x$

Domain: $|x| \ge 1$ | Range: $[0, \pi] \setminus \{\frac{\pi}{2}\}$

$y = \text{cosec}^{-1}x$

Domain: $|x| \ge 1$ | Range: $[-\frac{\pi}{2}, \frac{\pi}{2}] \setminus \{0\}$

Topic 3: Fundamental Properties & Symmetries

Property Group 1: Self-Adjusting Symmetries

These relations let you cancel a function with its inverse, but they only apply when the variables remain within their specific domain/range limits. Watch out for domain checks in board exams and JEE Mains!

$\sin^{-1}(\sin \theta) = \theta \quad \text{iff} \quad \theta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ (and $\sin(\sin^{-1} x) = x$ for $x \in [-1, 1]$)
$\cos^{-1}(\cos \theta) = \theta \quad \text{iff} \quad \theta \in [0, \pi]$ (and $\cos(\cos^{-1} x) = x$ for $x \in [-1, 1]$)
$\tan^{-1}(\tan \theta) = \theta \quad \text{iff} \quad \theta \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ (and $\tan(\tan^{-1} x) = x$ for $x \in \mathbb{R}$)
$\cot^{-1}(\cot \theta) = \theta \quad \text{iff} \quad \theta \in (0, \pi)$ (and $\cot(\cot^{-1} x) = x$ for $x \in \mathbb{R}$)
$\sec^{-1}(\sec \theta) = \theta \quad \text{iff} \quad \theta \in [0, \pi] \setminus \left\{\frac{\pi}{2}\right\}$ (and $\sec(\sec^{-1} x) = x$ for $|x| \ge 1$)
$\text{cosec}^{-1}(\text{cosec } \theta) = \theta \quad \text{iff} \quad \theta \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right] \setminus \{0\}$ (and $\text{cosec}(\text{cosec}^{-1} x) = x$ for $|x| \ge 1$)

Property Group 2: Reciprocal Relationships

Reciprocal angles require special attention for negative values of $x$. In particular, $\cot^{-1}x$ shifts by $\pi$ when $x < 0$ (a classic trap in JEE!):

$\sin^{-1}\left(\dfrac{1}{x}\right) = \text{cosec}^{-1}x \quad \text{for } |x| \ge 1$
$\cos^{-1}\left(\dfrac{1}{x}\right) = \sec^{-1}x \quad \text{for } |x| \ge 1$
$\tan^{-1}\left(\dfrac{1}{x}\right) = \begin{cases} \cot^{-1}x & \text{if } x > 0 \\ -\pi + \cot^{-1}x & \text{if } x < 0 \end{cases}$

Property Group 3: Even/Odd & Negative Input Symmetries

Negative arguments split cleanly into two groups based on symmetry:

Symmetric about Origin (Odd-like):

$\sin^{-1}(-x) = -\sin^{-1}x$
$\tan^{-1}(-x) = -\tan^{-1}x$
$\text{cosec}^{-1}(-x) = -\text{cosec}^{-1}x$

Shifted by $\pi$ (Reflected):

$\cos^{-1}(-x) = \pi - \cos^{-1}x$
$\cot^{-1}(-x) = \pi - \cot^{-1}x$
$\sec^{-1}(-x) = \pi - \sec^{-1}x$

Property Group 4: Complementary Co-Function Identities

The sum of complementary inverse trigonometric functions is always constant, matching the domain constraints of the functions:

$\sin^{-1}x + \cos^{-1}x = \dfrac{\pi}{2} \quad \text{for } x \in [-1, 1]$
$\tan^{-1}x + \cot^{-1}x = \dfrac{\pi}{2} \quad \text{for } x \in \mathbb{R}$
$\sec^{-1}x + \text{cosec}^{-1}x = \dfrac{\pi}{2} \quad \text{for } |x| \ge 1$

Topic 4: Advanced Composition & Algebra Formulas

Inverse Tangent Addition & Subtraction

Adding inverse tangents relies on the sign of $xy$ to avoid stepping outside the principal boundaries of $(-\pi/2, \pi/2)$:

$\tan^{-1}x + \tan^{-1}y = \tan^{-1}\left(\dfrac{x+y}{1-xy}\right) \quad \text{if } xy < 1$
$\tan^{-1}x + \tan^{-1}y = \pi + \tan^{-1}\left(\dfrac{x+y}{1-xy}\right) \quad \text{if } x > 0, y > 0, xy > 1$
$\tan^{-1}x + \tan^{-1}y = -\pi + \tan^{-1}\left(\dfrac{x+y}{1-xy}\right) \quad \text{if } x < 0, y < 0, xy > 1$
$\tan^{-1}x - \tan^{-1}y = \tan^{-1}\left(\dfrac{x-y}{1+xy}\right) \quad \text{if } xy > -1$

Inverse Sine & Cosine Sum/Difference

These identities are highly useful for proofs and algebraic combinations in R.S. Aggarwal exercises:

$\sin^{-1}x \pm \sin^{-1}y = \sin^{-1}\left(x\sqrt{1-y^2} \pm y\sqrt{1-x^2}\right) \quad \text{for } x^2+y^2 \le 1$
$\cos^{-1}x \pm \cos^{-1}y = \cos^{-1}\left(xy \mp \sqrt{1-x^2}\sqrt{1-y^2}\right) \quad \text{for } x,y \in [-1, 1]$

Multiple Angle Conversions

These conversion formulas are essential for simplifying algebraic equations and resolving calculus derivatives/integrals:

1. Doubled Inverse Tangent ($2\tan^{-1}x$): $$2\tan^{-1}x = \sin^{-1}\left(\frac{2x}{1+x^2}\right) \quad \text{for } |x| \le 1$$ $$2\tan^{-1}x = \cos^{-1}\left(\frac{1-x^2}{1+x^2}\right) \quad \text{for } x \ge 0$$ $$2\tan^{-1}x = \tan^{-1}\left(\frac{2x}{1-x^2}\right) \quad \text{for } -1 < x < 1$$

2. Tripled Angle Formulas:

$3\sin^{-1}x = \sin^{-1}(3x - 4x^3) \quad \text{for } -\dfrac{1}{2} \le x \le \dfrac{1}{2}$
$3\cos^{-1}x = \cos^{-1}(4x^3 - 3x) \quad \text{for } \dfrac{1}{2} \le x \le 1$
$3\tan^{-1}x = \tan^{-1}\left(\dfrac{3x - x^3}{1 - 3x^2}\right) \quad \text{for } -\dfrac{1}{\sqrt{3}} < x < \dfrac{1}{\sqrt{3}}$

Topic 5: Algebraic Simplification & Substitution Matrix

Use these standard trigonometric substitutions to simplify algebraic expressions. They eliminate square roots and convert expressions into single trigonometric terms.

Expression Pattern	Standard Substitution & Target Form
$a^2 - x^2$ or $\sqrt{a^2 - x^2}$	$x = a\sin\theta$ or $x = a\cos\theta$ $\implies a^2 - a^2\sin^2\theta = a^2\cos^2\theta$
$a^2 + x^2$ or $\sqrt{a^2 + x^2}$	$x = a\tan\theta$ or $x = a\cot\theta$ $\implies a^2 + a^2\tan^2\theta = a^2\sec^2\theta$
$x^2 - a^2$ or $\sqrt{x^2 - a^2}$	$x = a\sec\theta$ or $x = a\text{cosec}\theta$ $\implies a^2\sec^2\theta - a^2 = a^2\tan^2\theta$
$\sqrt{\dfrac{a-x}{a+x}}$ or $\sqrt{\dfrac{a+x}{a-x}}$	$x = a\cos\theta$ or $x = a\cos 2\theta$ $\implies \dfrac{a(1-\cos\theta)}{a(1+\cos\theta)} = \tan^2\left(\frac{\theta}{2}\right)$
$a\cos x \pm b\sin x$	Put $a = r\cos\alpha$ and $b = r\sin\alpha$ $\implies r = \sqrt{a^2+b^2}$ and $\alpha = \tan^{-1}(b/a)$

High-Yield R.S. Aggarwal Simplification Cheat Sheet

Memorize these standard algebraic-to-trigonometric simplifications to solve complicated differentiation, integration, and ITF equations rapidly:

$\tan^{-1}\left(\dfrac{\cos x - \sin x}{\cos x + \sin x}\right) = \dfrac{\pi}{4} - x$
$\tan^{-1}\left(\dfrac{\cos x + \sin x}{\cos x - \sin x}\right) = \dfrac{\pi}{4} + x$
$\tan^{-1}(\sec x + \tan x) = \dfrac{\pi}{4} + \dfrac{x}{2}$
$\tan^{-1}(\sec x - \tan x) = \dfrac{\pi}{4} - \dfrac{x}{2}$

$\cot^{-1}(\sec x + \tan x) = \dfrac{\pi}{4} - \dfrac{x}{2}$
$\tan^{-1}\left(\sqrt{\dfrac{1-\cos x}{1+\cos x}}\right) = \dfrac{x}{2}$
$\tan^{-1}\left(\sqrt{\dfrac{1+\cos x}{1-\cos x}}\right) = \dfrac{\pi}{2} - \dfrac{x}{2}$
$\tan^{-1}\left(\dfrac{x}{a + \sqrt{a^2-x^2}}\right) = \dfrac{1}{2}\sin^{-1}\left(\dfrac{x}{a}\right)$

Topic 6: Class 11 Trigonometry Prerequisite Vault

Class 11 Formula Vault

Inverse trigonometric algebraic steps rely heavily on Class 11 trigonometric identities. Review these core formulas to simplify expressions, substitute terms, and solve calculus problems.

1. Basic Pythagorean Identities

$\sin^2\theta + \cos^2\theta = 1$

$\sec^2\theta - \tan^2\theta = 1$

$\text{cosec}^2\theta - \cot^2\theta = 1$

2. Allied Angles & Quadrant Rules (ASTC Rule)

Symmetry Sines:

$\sin(-\theta) = -\sin\theta$
$\cos(-\theta) = \cos\theta$
$\tan(-\theta) = -\tan\theta$

Co-Function / Allied Shifts:

$\sin\left(\dfrac{\pi}{2} - \theta\right) = \cos\theta \quad | \quad \sin\left(\dfrac{\pi}{2} + \theta\right) = \cos\theta$
$\cos\left(\dfrac{\pi}{2} - \theta\right) = \sin\theta \quad | \quad \cos\left(\dfrac{\pi}{2} + \theta\right) = -\sin\theta$
$\sin(\pi - \theta) = \sin\theta \quad | \quad \cos(\pi - \theta) = -\cos\theta$

3. Compound Angle Identities

$\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B$
$\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B$

$\tan(A \pm B) = \dfrac{\tan A \pm \tan B}{1 \mp \tan A\tan B}$
$\cot(A \pm B) = \dfrac{\cot A\cot B \mp 1}{\cot B \pm \cot A}$

4. Double Angle & Linearizing Sub-Multiples

Double Angles:

$\sin 2\theta = 2\sin\theta\cos\theta = \dfrac{2\tan\theta}{1+\tan^2\theta}$
$\cos 2\theta = \cos^2\theta - \sin^2\theta = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta$
$\cos 2\theta = \dfrac{1-\tan^2\theta}{1+\tan^2\theta}$
$\tan 2\theta = \dfrac{2\tan\theta}{1-\tan^2\theta}$

Linearizing Half-Angles (Highly Used!):

$1 - \cos\theta = 2\sin^2\left(\dfrac{\theta}{2}\right)$
$1 + \cos\theta = 2\cos^2\left(\dfrac{\theta}{2}\right)$
$\dfrac{1-\cos\theta}{1+\cos\theta} = \tan^2\left(\dfrac{\theta}{2}\right)$
$\sin\theta = 2\sin\left(\dfrac{\theta}{2}\right)\cos\left(\dfrac{\theta}{2}\right)$

5. Triple Angle Formulas

$\sin 3\theta = 3\sin\theta - 4\sin^3\theta$

$\cos 3\theta = 4\cos^3\theta - 3\cos\theta$

$\tan 3\theta = \dfrac{3\tan\theta - \tan^3\theta}{1 - 3\tan^2\theta}$

6. Product-to-Sum & Sum-to-Product (C-D) Formulas

Sum to Product (C-D):

$\sin C + \sin D = 2\sin\left(\dfrac{C+D}{2}\right)\cos\left(\dfrac{C-D}{2}\right)$
$\sin C - \sin D = 2\cos\left(\dfrac{C+D}{2}\right)\sin\left(\dfrac{C-D}{2}\right)$
$\cos C + \cos D = 2\cos\left(\dfrac{C+D}{2}\right)\cos\left(\dfrac{C-D}{2}\right)$
$\cos C - \cos D = 2\sin\left(\dfrac{C+D}{2}\right)\sin\left(\dfrac{D-C}{2}\right)$

Product to Sum:

$2\sin A\cos B = \sin(A+B) + \sin(A-B)$
$2\cos A\sin B = \sin(A+B) - \sin(A-B)$
$2\cos A\cos B = \cos(A+B) + \cos(A-B)$
$2\sin A\sin B = \cos(A-B) - \cos(A+B)$

7. Frequently Used Specific Values

$\sin 15^\circ = \cos 75^\circ = \dfrac{\sqrt{3}-1}{2\sqrt{2}}$
$\cos 15^\circ = \sin 75^\circ = \dfrac{\sqrt{3}+1}{2\sqrt{2}}$
$\tan 15^\circ = 2 - \sqrt{3} \quad | \quad \cot 15^\circ = 2 + \sqrt{3}$

$\sin 18^\circ = \cos 72^\circ = \dfrac{\sqrt{5}-1}{4}$
$\cos 36^\circ = \sin 54^\circ = \dfrac{\sqrt{5}+1}{4}$

Topic 7: Last-Minute Recall Sheet

Quick Revision Points

Principal Angles: The output range for $\sin^{-1}x$, $\tan^{-1}x$, and $\text{cosec}^{-1}x$ is restricted to **Quadrant I and IV** ($[-\pi/2, \pi/2]$). The range for $\cos^{-1}x$, $\cot^{-1}x$, and $\sec^{-1}x$ is restricted to **Quadrant I and II** ($[0, \pi]$).
Negative Inputs: Swap negative inputs instantly using $\cos^{-1}(-x) = \pi - \cos^{-1}x$ and $\sin^{-1}(-x) = -\sin^{-1}x$.
Equations: When solving inverse equations (especially with tangents), **always plug the roots back** into the original equation to identify and discard extraneous solutions.
Calculus Setup: Use substitution patterns (like $x = a\tan\theta$ for $a^2+x^2$) to simplify complicated functions before taking their derivatives or integrals.