PREPWISER.
Practise online

Inverse Trigonometric Functions revision notes

Every idea in the Inverse Trigonometric Functions chapter for JEE Main, with the usual mistakes and the fast methods. Topic 1 is open to everyone; sign in free to read all 17 and download the PDF.

Sign in to read all 17Download PDF

Read the first part free. Sign in free with Google to read the rest and download the PDF: 9 A4 pages.9 A4 pages, ready to print.

How we count: 7× asked means 7 past JEE Main questions in the PrepWiser bank needed that concept, and "last" is the latest year one appeared. ★ marks ideas asked 8 or more times.

Principal values, domains and graphs

sin⁻¹, cos⁻¹, tan⁻¹: principal branches
3× asked, last 2023
sin⁡−1:[−1,1]→[−π2,π2],cos⁡−1:[−1,1]→[0,π],tan⁡−1:R→(−π2,π2)\sin^{-1}:[-1,1]\to\left[-\tfrac{\pi}{2},\tfrac{\pi}{2}\right],\qquad \cos^{-1}:[-1,1]\to[0,\pi],\qquad \tan^{-1}:\mathbb{R}\to\left(-\tfrac{\pi}{2},\tfrac{\pi}{2}\right)
Principal values only: the one angle in that range. sin⁡−1(1/2)=π/6\sin^{-1}(1/2)=\pi/6, cos⁡−1(−1/2)=2π/3\cos^{-1}(-1/2)=2\pi/3, tan⁡−1(−1)=−π/4\tan^{-1}(-1)=-\pi/4.
xxyy−1-11π2\frac\pi2−π2-\frac\pi2sin⁡−1x\sin^{-1}x
[−1,1]→[−π2,π2][-1,1]\to[-\frac\pi2,\frac\pi2]
xxyy−1-11π2\frac\pi2π\picos⁡−1x\cos^{-1}x
[−1,1]→[0,π][-1,1]\to[0,\pi]
xxyytan⁡−1x\tan^{-1}xπ2\frac\pi2−π2-\frac\pi2
R→(−π2,π2)\mathbb R\to(-\frac\pi2,\frac\pi2), never reaches ±π2\pm\frac\pi2
cot⁻¹, sec⁻¹, cosec⁻¹: principal branches
6× asked, last 2025
cot⁡−1:R→(0,π),sec⁡−1:∣x∣≥1→[0,π]−{π2},csc⁡−1:∣x∣≥1→[−π2,π2]−{0}\cot^{-1}:\mathbb{R}\to(0,\pi),\qquad \sec^{-1}:|x|\ge1\to[0,\pi]-\left\{\tfrac{\pi}{2}\right\},\qquad \csc^{-1}:|x|\ge1\to\left[-\tfrac{\pi}{2},\tfrac{\pi}{2}\right]-\{0\}
For ∣x∣≥1|x|\ge1: sec⁡−1x=cos⁡−1(1/x)\sec^{-1}x=\cos^{-1}(1/x) and csc⁡−1x=sin⁡−1(1/x)\csc^{-1}x=\sin^{-1}(1/x). cot⁡−1x\cot^{-1}x is never negative.
xxyycot⁡−1x\cot^{-1}xπ2\frac\pi2π\pi
R→(0,π)\mathbb R\to(0,\pi): never negative
xxyy−1-11sec⁡−1x\sec^{-1}xπ2\frac\pi2π\pi
∣x∣≥1→[0,π]−{π2}|x|\ge1\to[0,\pi]-\{\frac\pi2\}
xxyy−1-11π2\frac\pi2cosec⁡−1x\operatorname{cosec}^{-1}x−π2-\frac\pi2
∣x∣≥1→[−π2,π2]−{0}|x|\ge1\to[-\frac\pi2,\frac\pi2]-\{0\}
  • Not (−π/2,π/2](-\pi/2,\pi/2] for cot⁡−1\cot^{-1}. Its range is (0,π)(0,\pi), so a negative argument gives an obtuse angle: cot⁡−1(−1)=3π/4\cot^{-1}(-1)=3\pi/4, not −π/4-\pi/4.
Graphs and monotonicity
4× asked, last 2023
sin⁡−1, tan⁡−1: increasing, odd,cos⁡−1, cot⁡−1: decreasing\sin^{-1},\ \tan^{-1}:\ \text{increasing, odd},\qquad \cos^{-1},\ \cot^{-1}:\ \text{decreasing}
Each graph is its principal branch reflected in y=xy=x. Asymptotes: y=±π/2y=\pm\pi/2 for tan⁡−1\tan^{-1}; y=0y=0, y=πy=\pi for cot⁡−1\cot^{-1}.
xxyy−1-111−1-1sin⁡−1x\sin^{-1}xsin⁡x\sin xy=xy=x
sin⁡x\sin x on [−π2,π2][-\frac\pi2,\frac\pi2] (one-one) and its inverse: mirror images in y=xy=x
  • Quick method. Range of f(g(x))f(g(x)) with ff monotonic: find the range of gg, then map its ends, keeping open ends open. x2/(x2+1)x^2/(x^2+1) takes [0,1)[0,1), so 4sin⁡−14\sin^{-1} of it takes [0,2π)[0,2\pi).

Practise Inverse Trigonometric Functions questions as you read. Free, and you can start without signing up.

Practise Inverse Trigonometric Functions

14 more concepts in 6 topics

Domains of composite functions; Properties of inverse functions; Sum, difference and double-angle formulas; Simplifying inverse trig expressions; and Equations, series and extreme values, plus ideas from differentiation. Read the first part free. Sign in free with Google to read the rest and download the PDF.

Sign in to read the restYou come straight back to this spot.Download the PDF

Domains of composite functions

  • Domain of sin⁻¹(g(x)) and cos⁻¹(g(x))19×

Properties of inverse functions

  • Function of its inverse and back3×
  • Bringing the angle into the branch5×
  • Negative arguments4×
  • Complementary pairs8×
  • Reciprocal arguments6×

Sum, difference and double-angle formulas

  • tan⁻¹x ± tan⁻¹y12×
  • Double- and triple-angle forms10×
  • sin⁻¹x + sin⁻¹y, cos⁻¹x + cos⁻¹y7×

Simplifying inverse trig expressions

  • Trig ratio of an inverse trig angle14×
  • Simplifying by substitution14×

Equations, series and extreme values

  • Solving inverse trig equations13×
  • Telescoping series8×
  • Max and min with complementary pairs6×

Also asked here: differentiation

  • Key points1

Practise Inverse Trigonometric Functions next

About 95 past JEE Main questions from this chapter, with instant answer checking and step-by-step solutions where available. Free, and you can start without signing up.

Practise Inverse Trigonometric Functions

Also free on PrepWiser

Spotted a mistake? Write to support@prepwiser.in and we will fix it in the next version.

Version 1, October 2026. Free to share for study, as it is. Please don't sell it or remove the PrepWiser name.

PractiseSign in to read all 17Download PDF
Free with Google, takes a few seconds9 pages, free