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Schooling · Class 12 · Mathematics · Inverse Trigonometric Functions · Chapter on a page

CBSE Class 12 · Mathematics · NCERT Mathematics Part-I

Inverse Trigonometric Functions: chapter on a page

A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).

About the notes: Written by Shishya's AI on 27 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.

Fact-checked on 21 Oct 2026: an AI-assisted check of this note against cited sources; the errors it found were corrected that day. Some errors may remain.

Mind map

Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.

Mind map of Inverse Trigonometric Functions: Key ideas, Formulas and facts, Watch out forKey ideasA function must be one-to-one to…The principal-value branch of sin⁻¹…The principal-value branch of cos⁻¹…The principal-value branch of tan⁻¹…For cot⁻¹, cosec⁻¹ and sec⁻¹, the…Identities such as sin⁻¹ x + cos⁻¹ x…Sum and difference formulas, for…Formulas and factssin⁻¹ xcos⁻¹ xtan⁻¹ xcot⁻¹ xsec⁻¹ xcosec⁻¹ xComplementary identities (for x in…Watch out forIgnoring the restricted range and…Using tan⁻¹ x + tan⁻¹ y formula when…Confusing sin⁻¹ x with 1/sin xApplying sin⁻¹(−x) = π − sin⁻¹ x…Forgetting domain restrictions when…InverseTrigonometricFunctions
The mind map as a list
  • Key ideas
    • A function must be one-to-one to…
    • The principal-value branch of sin⁻¹…
    • The principal-value branch of cos⁻¹…
    • The principal-value branch of tan⁻¹…
    • For cot⁻¹, cosec⁻¹ and sec⁻¹, the…
    • Identities such as sin⁻¹ x + cos⁻¹ x…
    • Sum and difference formulas, for…
  • Formulas and facts
    • sin⁻¹ x
    • cos⁻¹ x
    • tan⁻¹ x
    • cot⁻¹ x
    • sec⁻¹ x
    • cosec⁻¹ x
    • Complementary identities (for x in…
  • Watch out for
    • Ignoring the restricted range and…
    • Using tan⁻¹ x + tan⁻¹ y formula when…
    • Confusing sin⁻¹ x with 1/sin x
    • Applying sin⁻¹(−x) = π − sin⁻¹ x…
    • Forgetting domain restrictions when…

What this chapter is about

In earlier classes you learned trigonometric functions such as sin, cos and tan that take an angle and return a ratio. Now you meet the reverse idea: given a ratio, find the angle. These are the inverse trigonometric functions, written sin⁻¹, cos⁻¹, tan⁻¹ and so on. Because the original functions repeat their values, we must restrict their domains to make them one-to-one before an inverse can exist. Understanding these restrictions is the heart of the chapter.

Key ideas

  • A function must be one-to-one to possess an inverse; trigonometric functions are made one-to-one by restricting their domains to principal-value branches.
  • The principal-value branch of sin⁻¹ has domain [−1, 1] and range [−π/2, π/2].
  • The principal-value branch of cos⁻¹ has domain [−1, 1] and range [0, π].
  • The principal-value branch of tan⁻¹ has domain (−∞, ∞) and range (−π/2, π/2).
  • For cot⁻¹, cosec⁻¹ and sec⁻¹, the domains exclude values where the original functions are undefined, and ranges are chosen to make them one-to-one.
  • Identities such as sin⁻¹ x + cos⁻¹ x = π/2 hold for every x in [−1, 1]; similar complementary relations exist for tan⁻¹ and cot⁻¹.
  • Sum and difference formulas, for instance tan⁻¹ x + tan⁻¹ y = tan⁻¹[(x + y)/(1 − xy)] when xy < 1, help simplify composite expressions.
  • Inverse trigonometric functions can be written in terms of each other, useful when converting an expression to a single function.

Formulas and facts to remember

  • Function: sin⁻¹ x · Domain: [−1, 1] · Principal-value range: [−π/2, π/2]
  • Function: cos⁻¹ x · Domain: [−1, 1] · Principal-value range: [0, π]
  • Function: tan⁻¹ x · Domain: (−∞, ∞) · Principal-value range: (−π/2, π/2)
  • Function: cot⁻¹ x · Domain: (−∞, ∞) · Principal-value range: (0, π)
  • Function: sec⁻¹ x · Domain: (−∞, −1] ∪ [1, ∞) · Principal-value range: [0, π], excluding π/2
  • Function: cosec⁻¹ x · Domain: (−∞, −1] ∪ [1, ∞) · Principal-value range: [−π/2, π/2], excluding 0
  • Complementary identities (for x in the respective domains):
  • sin⁻¹ x + cos⁻¹ x = π/2
  • tan⁻¹ x + cot⁻¹ x = π/2
  • sec⁻¹ x + cosec⁻¹ x = π/2
  • Negative-argument relations:
  • sin⁻¹(−x) = −sin⁻¹ x
  • cos⁻¹(−x) = π − cos⁻¹ x
  • tan⁻¹(−x) = −tan⁻¹ x
  • Sum formula for tan⁻¹ (when xy < 1):
  • tan⁻¹ x + tan⁻¹ y = tan⁻¹[(x + y)/(1 − xy)]
  • Double-angle form:
  • 2 tan⁻¹ x = sin⁻¹[2x/(1 + x²)] = cos⁻¹[(1 − x²)/(1 + x²)] = tan⁻¹[2x/(1 − x²)], valid for |x| ≤ 1 (sin⁻¹ form), x ≥ 0 (cos⁻¹ form) and −1 < x < 1 (tan⁻¹ form).

Common mistakes

  • Ignoring the restricted range and giving multiple angles → always pick the angle inside the principal-value branch.
  • Using tan⁻¹ x + tan⁻¹ y formula when xy ≥ 1 without adjusting by ±π → check the condition xy < 1 first; add or subtract π when needed.
  • Confusing sin⁻¹ x with 1/sin x → sin⁻¹ x is the inverse function (arc sine), not the reciprocal; write 1/sin x as cosec x.
  • Applying sin⁻¹(−x) = π − sin⁻¹ x, which holds for cos⁻¹, to sine → for sine and tangent, use the odd-function rule: sin⁻¹(−x) = −sin⁻¹ x.
  • Forgetting domain restrictions when simplifying expressions like sin(sin⁻¹ x) → such identities hold only when x is in the correct domain.

Quick revision

  • sin⁻¹ x lives in [−π/2, π/2]; cos⁻¹ x lives in [0, π]; tan⁻¹ x lives in (−π/2, π/2).
  • sin⁻¹ x + cos⁻¹ x = π/2 for all x in [−1, 1].
  • tan⁻¹ a + tan⁻¹ b = tan⁻¹[(a + b)/(1 − ab)] when ab < 1.
  • Odd-function rule: sin⁻¹(−x) = −sin⁻¹ x; tan⁻¹(−x) = −tan⁻¹ x.
  • Even-style rule for cosine: cos⁻¹(−x) = π − cos⁻¹ x.
  • Always verify the answer lies in the principal-value range.