Inverse trigonometry

This chapter rigorously introduces inverse trigonometric functions, establishing their principal values and fundamental identities. Mastery of these concepts is critical for solving complex problems and forms a foundational component for advanced topics in trigonometry and complex numbers, frequently appearing in CMI examinations.

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Chapter Contents

| Topic |

|---|-------| | 1 | Principal values | | 2 | Standard identities | | 3 | Composite inverse-trig expressions | | 4 | Equation solving with inverse trig |

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We begin with Principal values.

Part 1: Principal values

Principal Values

Overview

Principal values are the single chosen outputs of inverse trigonometric functions. Because trigonometric functions are periodic and not one-to-one on all of

\mathbb{R}

, inverse trig is defined by restricting to a standard range. Many exam questions in inverse trigonometry are really questions about identifying the correct principal value, not about raw trig manipulation. ---

Learning Objectives

❗ By the End of This Topic

After studying this topic, you will be able to:

State the principal-value ranges of $\sin^{-1}x$ , $\cos^{-1}x$ , and $\tan^{-1}x$ .

Compute principal values such as $\sin^{-1}(\sin \theta)$ and $\cos^{-1}(\cos \theta)$ .

Reduce angles to the correct principal-value interval.

Use piecewise reasoning for principal-value expressions.

Avoid the false rule that inverse trig simply cancels trig for all angles.

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Principal Value Ranges

📐 Standard Ranges

$\sin^{-1}x \in \left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]$

$\cos^{-1}x \in [0,\pi]$

$\tan^{-1}x \in \left(-\dfrac{\pi}{2},\dfrac{\pi}{2}\right)$

These are not optional conventions in a given problem — they define the principal value. ---

Why Principal Values Are Needed

📖 Why Inverse Trig Is Restricted

The equation
$\qquad \sin\theta = y$
has infinitely many solutions for most $y$ .

So $\sin^{-1}y$ is defined to mean the unique solution in the interval
$\qquad \left[-\dfrac{\pi}{2},\dfrac{\pi}{2}\right]$ .

The same idea applies to $\cos^{-1}$ and $\tan^{-1}$ .

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Standard Principal Value Forms

📐 For

\sin^{-1}(\sin x)

[-\pi,\pi]

\sin^{-1}(\sin x)= \begin{cases}-\pi-x, & -\pi \le x < -\dfrac{\pi}{2}\

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Inverse trigonometry

Inverse trigonometry

Chapter Contents

| Topic |

Part 1: Principal values

Principal Values

Overview

Learning Objectives

Principal Value Ranges

Why Principal Values Are Needed

Standard Principal Value Forms

Minimal Worked Examples

How to Compute Principal Values

Geometry of Reflection Back into the Range

Common Patterns

Common Mistakes

CMI Strategy

Practice Questions

Summary

Part 2: Standard identities

Standard Identities

Overview

Learning Objectives

Principal Value Ranges

Core Standard Identities

Sum Formula for tan⁡−1\tan^{-1}tan−1

Very Useful Special Cases

Standard Derived Forms

Minimal Worked Examples

Common Mistakes

CMI Strategy

Practice Questions

Summary

Part 3: Composite inverse-trig expressions

Composite Inverse-Trig Expressions

Overview

Learning Objectives

Principal Ranges You Must Remember

Direct Compositions

Reverse-Type Compositions

Standard Composite Simplifications

Why the Square Root Is Positive

Triangle Method

Minimal Worked Examples

Common Patterns

Common Mistakes

CMI Strategy

Practice Questions

Summary

Part 4: Equation solving with inverse trig

Equation Solving with Inverse Trig

Overview

Learning Objectives

Principal Value Facts

Standard One-Step Equations

Equality of Inverse-Trig Functions

Very Important Identity

Solving Sum Equations

Minimal Worked Examples

Domain Checks You Must Not Skip

Common Patterns

Common Mistakes

CMI Strategy

Practice Questions

Summary

Chapter Summary

Chapter Review Questions

What's Next?

🎯 Key Points to Remember

Related Topics in Trigonometry and Complex Numbers

Geometry of complex numbers

Trigonometric equations

Polar form

Trigonometric transformations

More Resources

Study Notes

Short Notes

Test Series

Mock Tests

Previous Year Papers

Sum Formula for $\tan^{-1}$