Introduction

Indirect measurement — finding a length or an angle without physically measuring it — is one of the oldest applications of trigonometry, and inverse trigonometric functions are the tool that makes it possible whenever the UNKNOWN quantity is an angle rather than a side. Trigonometric functions and their inverses are widely used in engineering and in other sciences including physics, and beyond solving triangles, they also let us evaluate certain families of integrals in calculus.

Three motivating pictures.

• Slope problem. A line $y=mx+b$ makes an angle $\theta$ with the $x$-axis. Since slope is $m=\Delta y/\Delta x=\tan\theta$, recovering $\theta$ from the known slope $m$ needs the inverse tangent function.
• Movie-theatre viewing angle. A screen $7$ m tall has its bottom $2$ m above eye level; a viewer sits $x$ m from the screen. The viewing angle is $\theta(x)=\tan^{-1}\left(\dfrac9x\right)-\tan^{-1}\left(\dfrac2x\right)$ — a genuine function of $x$, built entirely from inverse tangents.
• Drawbridge. Two $40$ m leaves of a drawbridge must open wide enough to let a $33$ m-wide ship through; finding the minimum opening angle again needs an inverse trigonometric function.

Inverse trigonometric functions also show up inside calculus, in integrals such as $\displaystyle\int\dfrac{dx}{\sqrt{a^2-x^2}}$ and $\displaystyle\int\dfrac{dx}{a^2+x^2}$, whose antiderivatives are $\sin^{-1}$ and $\tan^{-1}$ expressions respectively — a preview of why this chapter matters well beyond angle-finding.

A more everyday picture. An oscilloscope — the electronic instrument that turns an electrical signal into a graph shaped like a sine curve, whose controls let you change that curve's amplitude, period and phase shift — is used, among other things, to measure human heartbeats; trigonometric functions play the dominant role in reading a trace like that.

Chapter roadmap. Building on Class XI's study of the six trigonometric functions of a real number (periodicity, domain, range), this chapter (i) defines each inverse trigonometric function, (ii) draws its graph, and (iii) develops and applies its properties to evaluate expressions and solve equations. Throughout, $\mathbb{R}$ denotes the real numbers and $\mathbb{Z}$ the integers.