Chapter 2: Inverse Trigonometric Functions

The trigonometric functions $\sin, \cos, \tan, \cot, \sec, \csc$ are not one-one on their natural domains , $\sin$ repeats every $2\pi$, $\tan$ every $\pi$. To invert them, we restrict each to a maximal interval on which it is monotonic; the inverse on that interval is the principal value branch. The chapter is short on theory and dense on identities. You will memorise about a dozen relations, each derived from the corresponding trigonometric identity, and use them to simplify expressions, solve equations, and prepare for the integrals chapter, where many anti-derivatives are inverse trigonometric.

The conceptual difficulty here is the same as in Chapter 1: a function admits an inverse only when restricted to a domain where it is one-one. The principal branches are chosen by convention to be near zero whenever possible , for instance $\arcsin$ outputs values in $[-\pi/2, \pi/2]$, not in $[\pi/2, 3\pi/2]$. Reading the convention correctly is essential for assigning the right sign and quadrant in problems.

Board questions in this chapter are usually computational: simplify $\sin^{-1}(\sin \tfrac{5\pi}{4})$ or evaluate $\tan^{-1} 1 + \tan^{-1} 2 + \tan^{-1} 3$. JEE questions push the same identities further, often combined with quadratic equations or summations. The identities themselves are easy if you remember their derivation: each one comes from feeding a trigonometric identity through the inverse.

A practical tip: when you see $\sin^{-1}(\text{something})$, ask first which branch the answer lives in. The branch determines the sign. Many mistakes in this chapter come from forgetting the principal range.

The chapter ties to the rest of Class XII in two places. First, integration formulas such as $\int \dfrac{dx}{1 + x^2} = \arctan x + C$ rely on these functions being available. Second, in physics and engineering, every angle that appears as an output of a measurement is implicitly the inverse of a trigonometric function.

What's inside

1. Principal value branches , how we choose the domain to invert.
2. Graphs of inverse trigonometric functions , and how to read them.
3. Basic identities , $\sin^{-1} x + \cos^{-1} x$, $\tan^{-1} x + \cot^{-1} x$ and friends.
4. Addition formulas , $\sin^{-1} x \pm \sin^{-1} y$, $\tan^{-1} x \pm \tan^{-1} y$.
5. Multiple-angle identities , $2 \sin^{-1} x$, $2 \tan^{-1} x$, $3 \sin^{-1} x$.
6. Equations and applications , solving and simplifying.

Key results / Formula card

Core identities:

• $\sin^{-1} x + \cos^{-1} x = \pi/2$.
• $\tan^{-1} x + \cot^{-1} x = \pi/2$.
• $\sec^{-1} x + \csc^{-1} x = \pi/2$.
• $\sin^{-1}(-x) = -\sin^{-1} x$, $\cos^{-1}(-x) = \pi - \cos^{-1} x$.
• $\tan^{-1} x + \tan^{-1} y = \tan^{-1} \dfrac{x + y}{1 - xy}$ if $xy < 1$.
• $2 \tan^{-1} x = \tan^{-1} \dfrac{2x}{1 - x^2}$ if $|x| < 1$.
• $2 \tan^{-1} x = \sin^{-1} \dfrac{2x}{1 + x^2} = \cos^{-1} \dfrac{1 - x^2}{1 + x^2}$ (with care on signs).
• $\sin^{-1}(\sin x) = x$ if $x \in [-\pi/2, \pi/2]$; otherwise reduce to the principal branch.

How to read this chapter

Start with the principal value branches. Then memorise the basic identities. Every later identity is a corollary; deriving them once is enough to lock them in.