Class 12 Mathematics Chapter 2: Inverse Trigonometric Functions

CBSE 2026–27 | NCERT-aligned free resource

1. Why an Inverse?

Trigonometric functions are periodic, so they are not one-one on all real numbers. Restricting their domains gives principal-value branches on which inverse functions can be defined.

2. Principal Value Ranges

sin⁻¹x has range [−π/2,π/2]; cos⁻¹x has range [0,π]; tan⁻¹x has range (−π/2,π/2). Their domains are respectively [−1,1], [−1,1] and R.

3. Important Identities

Within their principal-value conventions: sin⁻¹x + cos⁻¹x = π/2 for x∈[−1,1]. Also tan⁻¹x + tan⁻¹y = tan⁻¹((x+y)/(1−xy)) when the appropriate principal-value condition is satisfied.

4. Graph Idea

The graph of an inverse function is the reflection of the restricted original graph in y=x.

Worked Example

sin⁻¹(1/2)=π/6 because π/6 lies in the principal range of sin⁻¹ and sin(π/6)=1/2.

Exam Traps

• sin⁻¹x means inverse sine, not 1/sin x.
• Always check the principal-value range.
• Do not apply tangent addition identities without checking the principal-value adjustment.

Practice

1. State domains and principal ranges of the inverse trigonometric functions.
2. Evaluate standard inverse-trigonometric values.
3. Prove identities involving sin⁻¹, cos⁻¹ and tan⁻¹.
4. Sketch a restricted trigonometric function and its inverse.

Deep-Dive Revision & Worked Examples

Principal Values

Inverse trigonometric functions are functions only after a restricted principal-value domain is selected. This restriction is essential when interpreting sin⁻¹, cos⁻¹ and tan⁻¹.

Worked Example

Evaluate sin⁻¹(1/2). The principal value lies in [−π/2,π/2], so the answer is π/6.

Identity Example

For an expression such as sin⁻¹x+cos⁻¹x, use the principal-value relationship sin⁻¹x+cos⁻¹x=π/2 for x∈[−1,1].

Graph & Range Visual

Common Exam Traps

• Confusing reciprocal notation with inverse-function notation.
• Ignoring principal-value restrictions.
• Applying an identity outside its valid domain.
• Dropping brackets around inverse trigonometric expressions.

Practice

1. Evaluate inverse-trigonometric expressions using principal values.
2. Prove standard inverse-trigonometric identities with correct domains.
3. Simplify expressions containing tan⁻¹ and cot⁻¹.
4. Determine the domain and range of a composite inverse-trigonometric expression.

Common Mistakes — Chapter 2

• Reading sin⁻¹x as 1/sinx; it denotes an inverse function, not a reciprocal.
• Ignoring principal-value ranges.
• Using an identity without checking its domain.