Chapter Hub
Inverse Trigonometric Functions
Inverse trigonometric functions are used to find angles when a trigonometric ratio is known. Since ordinary trigonometric functions are many-one on their natural domains, each inverse function is defined only after choosing a principal value branch. The chapter begins with domains and ranges of sin⁻¹x, cos⁻¹x, tan⁻¹x, cot⁻¹x, sec⁻¹x and cosec⁻¹x. These intervals are not optional details; they decide whether an answer such as π/6, 5π/6, −π/6 or 7π/6 is acceptable. A major exam focus is simplification using identities such as sin⁻¹x + cos⁻¹x = π/2 and tan⁻¹x + cot⁻¹x = π/2. You must check the domain of the variable and the range of the final inverse function before applying a formula. Sum and difference formulae, especially for tan⁻¹x, are useful in reducing expressions and solving equations. The condition xy < 1 or xy > 1 affects whether an additional π adjustment is needed, so branch checking is essential. Graph-based understanding helps you remember why branches are restricted. The graph of an inverse function is obtained by reflecting the restricted original function in the line y = x, with domain and range interchanged.
Difficulty
Medium
Study time
70-90 min
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First full pass
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Key Concepts
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All Concepts
All 5 concepts of the chapter, in the order the NCERT chapter teaches them.
Domain and Principal Value Branches of Inverse Trigonometric Functions
The principal value branch of an inverse trigonometric function is the chosen interval of angles on which the corresponding trigonometric function becomes one-one and onto its required range.
Graphs of Inverse Trigonometric Functions
The graph of an inverse trigonometric function is obtained by reflecting the graph of the corresponding restricted trigonometric function in the line y = x.
Identities and Properties of Inverse Trigonometric Functions
Inverse trigonometric identities are relations between inverse functions that hold under specified domain and branch conditions.
Sum and Difference Formulae for Inverse Trigonometric Functions
Sum and difference formulae combine two inverse trigonometric expressions into a single inverse expression, with conditions determined by the principal value branch.
Simplifying and Solving Equations with Inverse Trigonometric Functions
Simplification and equation solving with inverse trigonometric functions means reducing expressions or finding variable values while preserving domain restrictions and principal value conditions.
Exam Intelligence
The traps that cost marks in this chapter, and how its main concepts turn up in the paper.
Common Traps
- Choosing an angle outside the principal value range.
- Using a trigonometric equation solution set as the value of an inverse trigonometric function.
- Reflecting the full periodic graph instead of the restricted branch.
- Applying tan⁻¹ sum formula without checking xy and signs.
- Accepting algebraic candidates without substituting back into the original equation.
How CBSE Asks This Chapter
- Domain and Principal Value Branches of Inverse Trigonometric Functions
Questions usually ask for principal values, domains, ranges, or the validity of a simplification involving inverse functions.
- Graphs of Inverse Trigonometric Functions
Students may be asked to identify a graph, mark domain and range, label reflected points, or explain why a restricted branch is needed.
- Identities and Properties of Inverse Trigonometric Functions
Questions ask students to simplify expressions, prove identities, convert between inverse functions, or evaluate exact values using complementary relations.
- Sum and Difference Formulae for Inverse Trigonometric Functions
Exam questions ask for exact evaluation, proof of an identity, or reduction of multiple inverse tangent terms to a single angle.
Quick Revision
Concept, formula or equation to remember, and the trap that loses marks — in one scannable view.
Domain and Principal Value Branches of Inverse Trigonometric Functions
Remember
Domains and ranges: sin⁻¹x: domain [−1,1], range [−π/2,π/2]; cos⁻¹x: domain [−1,1], range [0,π]; tan⁻¹x: domain R, range (−π/2,π/2); cot⁻¹x: domain R, range (0,π); sec⁻¹x: domain (−∞,−1] ∪ [1,∞), range [0,π] excluding π/2; cosec⁻¹x: domain (−∞,−1] ∪ [1,∞), range [−π/2,π/2] excluding 0.Trap to avoid
Writing all possible angles instead of the principal value, such as giving sin⁻¹(1/2) = nπ + (−1)ⁿπ/6.
Graphs of Inverse Trigonometric Functions
Remember
Graph rule: if y = f⁻¹(x), then its graph is the reflection of y = f(x) in y = x, after f is restricted to a one-one principal branch.Trap to avoid
Reflecting the full sine or cosine graph without restricting it first.
Identities and Properties of Inverse Trigonometric Functions
Remember
Key identities: sin⁻¹x + cos⁻¹x = π/2 for x ∈ [−1,1]; tan⁻¹x + cot⁻¹x = π/2 for x ∈ R; sec⁻¹x + cosec⁻¹x = π/2 for |x| ≥ 1, with branch conventions respected.Trap to avoid
Using an identity outside its domain, such as applying sin⁻¹x + cos⁻¹x when x is not in [−1,1].
Sum and Difference Formulae for Inverse Trigonometric Functions
Remember
For real x,y with xy < 1: tan⁻¹x + tan⁻¹y = tan⁻¹((x+y)/(1−xy)).Trap to avoid
Applying tan⁻¹x + tan⁻¹y = tan⁻¹((x+y)/(1−xy)) without checking whether xy < 1, xy = 1 or xy > 1.
Simplifying and Solving Equations with Inverse Trigonometric Functions
Remember
Method rules: preserve domains such as x ∈ [−1,1] for sin⁻¹x and cos⁻¹x; use identities only under their conditions; when applying trig functions to both sides, check principal ranges; for equations involving tan⁻¹, verify denominator conditions in sum and difference formulae.Trap to avoid
Taking sine, cosine or tangent on both sides and accepting every algebraic solution without checking the original inverse trigonometric equation.
| Concept | Remember | Trap to avoid |
|---|---|---|
| Domain and Principal Value Branches of Inverse Trigonometric Functions | Domains and ranges: sin⁻¹x: domain [−1,1], range [−π/2,π/2]; cos⁻¹x: domain [−1,1], range [0,π]; tan⁻¹x: domain R, range (−π/2,π/2); cot⁻¹x: domain R, range (0,π); sec⁻¹x: domain (−∞,−1] ∪ [1,∞), range [0,π] excluding π/2; cosec⁻¹x: domain (−∞,−1] ∪ [1,∞), range [−π/2,π/2] excluding 0. | Writing all possible angles instead of the principal value, such as giving sin⁻¹(1/2) = nπ + (−1)ⁿπ/6. |
| Graphs of Inverse Trigonometric Functions | Graph rule: if y = f⁻¹(x), then its graph is the reflection of y = f(x) in y = x, after f is restricted to a one-one principal branch. | Reflecting the full sine or cosine graph without restricting it first. |
| Identities and Properties of Inverse Trigonometric Functions | Key identities: sin⁻¹x + cos⁻¹x = π/2 for x ∈ [−1,1]; tan⁻¹x + cot⁻¹x = π/2 for x ∈ R; sec⁻¹x + cosec⁻¹x = π/2 for |x| ≥ 1, with branch conventions respected. | Using an identity outside its domain, such as applying sin⁻¹x + cos⁻¹x when x is not in [−1,1]. |
| Sum and Difference Formulae for Inverse Trigonometric Functions | For real x,y with xy < 1: tan⁻¹x + tan⁻¹y = tan⁻¹((x+y)/(1−xy)). | Applying tan⁻¹x + tan⁻¹y = tan⁻¹((x+y)/(1−xy)) without checking whether xy < 1, xy = 1 or xy > 1. |
| Simplifying and Solving Equations with Inverse Trigonometric Functions | Method rules: preserve domains such as x ∈ [−1,1] for sin⁻¹x and cos⁻¹x; use identities only under their conditions; when applying trig functions to both sides, check principal ranges; for equations involving tan⁻¹, verify denominator conditions in sum and difference formulae. | Taking sine, cosine or tangent on both sides and accepting every algebraic solution without checking the original inverse trigonometric equation. |
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