NCERT Solutions for Class 12 Maths Chapter 2: Inverse Trigonometric Functions
Principal value branches of the six inverse trigonometric functions, evaluating principal values, and simplifying expressions by a trigonometric substitution. Our own step-by-step solutions to Exercises 2.1, 2.2 and the Miscellaneous Exercise, one page per exercise, set out for step marks.
3 exercises, 43 questions
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Chapter 2 exercises
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What it tests. The principal value is the one angle in the function's principal branch whose ratio is the given number. Principal value branches: \(\sin^{-1}\): \(\left[-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right]\), \(\cos^{-1}\): \([0, \pi]\), \(\tan^{-1}\): \(\left(-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right)\), \(\cot^{-1}\): \((0, \pi)\), \(\sec^{-1}\): \([0, \pi] - \left\{\tfrac{\pi}{2}\right\}\), \(\csc^{-1}\): \(\left[-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right] - \{0\}\). Negative inputs: \(\sin^{-1}(-x) = -\sin^{-1}x\) but \(\cos^{-1}(-x) = \pi - \cos^{-1}x\).
What it tests. Substitute \(x = \sin\theta\), \(\cos\theta\) or \(\tan\theta\) (or \(a\sin\theta\), \(a\tan\theta\)) so the expression inside becomes a single trigonometric ratio of a multiple of \(\theta\); then \(\sin^{-1}(\sin t) = t\) only when \(t\) lies in the principal branch, so always check where \(t\) lies. For \(\sin^{-1}(\sin t)\) with \(t\) outside the branch, move to the angle in the branch with the same sine.
What it tests. Mixed practice: reduce \(\cos^{-1}(\cos t)\) or \(\tan^{-1}(\tan t)\) to the principal branch, prove sums of inverse functions by working with one right triangle per term, and solve equations by taking \(\tan\) or \(\sin\) of both sides (then check the roots in the original equation).
Done the NCERT exercises? The board paper asks more
Inverse Trigonometric Functions has 39 original board-style questions (MCQ, assertion–reason, short and long answers, case studies) with step mark schemes, revision notes and a four-step Route to 95. Three sample questions are open to everyone; a free account opens the rest.
Textbook: NCERT Mathematics Class 12, Parts I and II (rationalised edition, 2023-24 reprint onward), free from ncert.nic.in. Question statements are shortened to the minimum needed; the solutions and tips are our own. CBSE Math Revision is independent and not affiliated with NCERT or CBSE. Spotted a slip? Tell us and it goes in the corrections log.