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Inverse Trigonometric Functions formula sheet

Every formula in the Inverse Trigonometric Functions chapter for JEE Main, with one line on when it applies. Topic 1 is open to everyone; sign in free to read all 18 and download the PDF.

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

Principal values, domains and graphs

sin⁻¹, cos⁻¹, tan⁻¹
3× asked, last 2023
sin⁡−1:[−1,1]→[−π2,π2],cos⁡−1:[−1,1]→[0,π]tan⁡−1:R→(−π2,π2)\begin{aligned}&\sin^{-1}:[-1,1]\to\left[-\tfrac\pi2,\tfrac\pi2\right],\qquad \cos^{-1}:[-1,1]\to[0,\pi]\\&\tan^{-1}:\mathbb R\to\left(-\tfrac\pi2,\tfrac\pi2\right)\end{aligned}
Principal values: 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.
cot⁻¹, sec⁻¹, cosec⁻¹
6× asked, last 2025
cot⁡−1:R→(0,π),sec⁡−1:∣x∣≥1→[0,π]−{π2}csc⁡−1:∣x∣≥1→[−π2,π2]−{0}\begin{aligned}&\cot^{-1}:\mathbb R\to(0,\pi),\qquad \sec^{-1}:|x|\ge1\to[0,\pi]-\left\{\tfrac\pi2\right\}\\&\csc^{-1}:|x|\ge1\to\left[-\tfrac\pi2,\tfrac\pi2\right]-\{0\}\end{aligned}
sec⁡−1x=cos⁡−1(1/x)\sec^{-1}x=\cos^{-1}(1/x), csc⁡−1x=sin⁡−1(1/x)\csc^{-1}x=\sin^{-1}(1/x). cot⁡−1(−1)=3π/4\cot^{-1}(-1)=3\pi/4 (never negative).
Monotonicity
4× asked, last 2023
sin⁡−1, tan⁡−1: increasing, oddcos⁡−1, cot⁡−1: decreasing\begin{aligned}&\sin^{-1},\ \tan^{-1}:\ \text{increasing, odd}\\&\cos^{-1},\ \cot^{-1}:\ \text{decreasing}\end{aligned}
Range of f(g(x))f(g(x)): range of gg, then map the ends (open stays open).
Domains of composites
19× asked, last 2026
sin⁡−1g, cos⁡−1g: g2≤1;sec⁡−1g, csc⁡−1g: g2≥1\sin^{-1}g,\ \cos^{-1}g:\ g^2\le1;\qquad \sec^{-1}g,\ \csc^{-1}g:\ g^2\ge1
Intersect with every other term. g=N/Dg=N/D: N2≤D2N^2\le D^2, D≠0D\ne0. sin⁡−1[g]\sin^{-1}[g]: −1≤g<2-1\le g<2.

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Practise Inverse Trigonometric Functions

14 more formulas in 3 topics

Properties of inverse functions; Sum, difference and double angles; and Simplifying and solving. Read the first part free. Sign in free with Google to read the rest and download the PDF.

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Properties of inverse functions

  • Inverse and back3×
  • Into the branch5×
  • Negative arguments4×
  • Complementary pairs8×
  • Reciprocal arguments6×

Sum, difference and double angles

  • tan⁻¹x ± tan⁻¹y12×
  • Double angle10×
  • Other multiples
  • sin⁻¹x + sin⁻¹y7×

Simplifying and solving

  • Ratio of an inverse angle14×
  • Substitution14×
  • Equations13×
  • Telescoping series8×
  • Max and min6×

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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

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

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