Chapter 2 · Introduction

Chapter 2 — Inverse Trigonometric Functions

Every angle has one sine, but every sine belongs to infinitely many angles. Restrict the trigonometric functions to principal value branches and they can be run backwards.

An engineer laying a mountain road knows the rise and the length of each stretch and needs its angle. A robot arm knows where its hand must go and must work out the angles of its joints. A satellite dish installer in Kochi knows the satellite's position and must set the dish's elevation. Each time the value of a sine, cosine or tangent is known and the angle is wanted — trigonometry run backwards. Calculators do it with the keys $\sin^{-1}$, $\cos^{-1}$ and $\tan^{-1}$.

There is a catch, and it is exactly the idea of Chapter 1. Sine takes the value $\tfrac12$ at $\tfrac\pi6$, at $\tfrac{5\pi}6$, at $\tfrac{13\pi}6$ and at infinitely many other angles, so “the angle whose sine is $\tfrac12$” is not yet a function. In this chapter we cut each trigonometric function down to a principal value branch on which it is one-one and onto, invert it there, draw the six inverse functions by reflecting in the line $y=x$, and learn to evaluate expressions like $\sin^{-1}\!\left(\sin\tfrac{2\pi}3\right)$ correctly — the answer is not $\tfrac{2\pi}3$. In Chapter 5 you will differentiate these functions, and in Chapter 7 they appear as answers to integrals.

sin, cos, tan are many-one→Restrict to a branch→One-one & onto→sin⁻¹, cos⁻¹, tan⁻¹ …→Principal values→Graphs by reflection→sin⁻¹(sin θ)

🎯 By the end of this chapter you can…

explain why sine, cosine and tangent must be restricted before they can be inverted, and state the domain and principal value branch of all six inverse trigonometric functions;
find principal values such as $\cos^{-1}\!\left(-\tfrac12\right)$, $\sec^{-1}(-2)$ and $\cot^{-1}(-1)$ without a calculator, and evaluate expressions built from them;
find the domain and range of functions such as $\sin^{-1}(2x-1)$ or $3\cos^{-1}x-\tfrac\pi2$;
sketch the graphs of the six inverse functions by reflecting in $y=x$, and read off their domains, ranges, asymptotes and monotonicity;
evaluate $\sin^{-1}(\sin\theta)$, $\cos^{-1}(\cos\theta)$, $\tan^{-1}(\tan\theta)$ for any angle $\theta$, and expressions such as $\sin\!\left(\cos^{-1}\tfrac35\right)$;
answer Board 1-, 2- and 3-mark questions and the JEE-style MCQs on this chapter.

Choose a value $v$. On the unit circle, the angles with $\sin\theta=v$ are where the dashed horizontal line $y=v$ meets the circle — two points, and every full turn adds more angles. The list shows all of them between $-2\pi$ and $2\pi$. A calculator must pick one: the red point on the thick purple arc. Switch to cosine to see a different arc chosen. Press Play to sweep $v$.

function:$v$:

$v$—

angles in $[-2\pi,2\pi]$—

the calculator's answer—

Sections in this chapter

2.1

Inverse Trigonometric Functions and Principal Value Branches

Why sine must be restricted; the six inverse functions, their domains and principal value branches; finding principal values.

$y=\sin^{-1}x\iff\sin y=x,\ |y|\le\tfrac\pi2$

Open section →2.2

Graphs and Principal Values

Graphs by reflection in $y=x$; asymptotes and monotonicity; evaluating $\sin^{-1}(\sin\theta)$ and its cousins for any angle.

$\sin^{-1}\!\left(\sin\tfrac{2\pi}3\right)=\tfrac\pi3$

Open section →

✅ Before you start

You need one-one, onto and bijective functions and the horizontal line test (Section 1.2). From Class 11 Mathematics, Ch. 3 — Trigonometric Functions: radian measure, the values of all six functions at $0,\tfrac\pi6,\tfrac\pi4,\tfrac\pi3,\tfrac\pi2$, their signs in the four quadrants, allied angles such as $\sin(\pi-\theta)=\sin\theta$ and $\cos(2\pi-\theta)=\cos\theta$, and the graphs of $\sin x$, $\cos x$ and $\tan x$. Quick self-test: without a calculator, give $\cos\tfrac{2\pi}3$, $\tan\tfrac{3\pi}4$ and $\sec\tfrac{5\pi}6$. (Answers: $-\tfrac12$, $-1$, $-\tfrac2{\sqrt3}$.)

← 1.2 Types of Functions 2.1 Inverse Trigonometric Functions and Principal Value Branches →