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Cheat sheet · JEE Preparation Mathematics

Inverse Trigonometric Functions cheat sheet

The whole chapter on one page: key formulas, facts and the points to remember, for quick revision before a test.

JEE · Mathematics

Cheat sheet

Class 12 · Chapter 2Inverse Trigonometric Functions

Principal ranges: sin⁻¹ [−π/2, π/2], cos⁻¹ [0, π], tan⁻¹ (−π/2, π/2).

1. Domains & Principal Ranges

FunctionDomainRange
sin⁡−1x\sin^{-1}x[−1, 1][−π2,π2][-\tfrac\pi2, \tfrac\pi2]
cos⁡−1x\cos^{-1}x[−1, 1][0,π][0, \pi]
tan⁡−1x\tan^{-1}xℝ(−π2,π2)(-\tfrac\pi2, \tfrac\pi2)
cot⁡−1x\cot^{-1}xℝ(0,π)(0, \pi)
sec⁡−1x\sec^{-1}x|x| ≥ 1[0,π]−{π2}[0, \pi] - \{\tfrac\pi2\}
csc⁡−1x\csc^{-1}x|x| ≥ 1[−π2,π2]−{0}[-\tfrac\pi2, \tfrac\pi2] - \{0\}

2. Properties

sin⁡−1(−x)=−sin⁡−1x\sin^{-1}(-x) = -\sin^{-1}xcos⁡−1(−x)=π−cos⁡−1x\cos^{-1}(-x) = \pi - \cos^{-1}x
sin⁡−1x+cos⁡−1x=π2\sin^{-1}x + \cos^{-1}x = \dfrac\pi2
tan⁡−1x+cot⁡−1x=π2\tan^{-1}x + \cot^{-1}x = \dfrac\pi2

3. Key Points

  • Always answer in the principal range.
  • sin⁻¹x ≠ 1/sin x.
  • Graph of inverse: reflect in y = x.
  • sin⁻¹(sin x) = x only in the principal range.

Remember: Domain & range table | complementary identities

Inverse Trigonometric Functions: questions and notes

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