Q1. Find the value of sin⁻¹(1/2).
We need θ in [−π/2, π/2] with sinθ = 1/2
θ = π/6 (i.e. 30°)
sin⁻¹(1/2) = π/6
Inverse Trigonometric Functions — Class 12
Trig functions are many-to-one, so their inverses need a restricted "principal value branch" to make sense. Learn the ranges cold.
Why restrict the range?sin, cos, tan repeat their values infinitely often, so their inverses are only defined by picking one standard (principal) output range.
Principal value branchessin⁻¹x: [−π/2, π/2]. cos⁻¹x: [0, π]. tan⁻¹x: (−π/2, π/2).
Key identitysin⁻¹x + cos⁻¹x = π/2 for all x in [−1, 1].
Solved examples
Q2. Find cos⁻¹(1/2) using the identity sin⁻¹x + cos⁻¹x = π/2.
sin⁻¹(1/2) = π/6 (from above)
cos⁻¹(1/2) = π/2 − π/6
= 3π/6 − π/6 = 2π/6 = π/3
Formula sheet
| Quantity | Formula |
|---|---|
| Domain of sin⁻¹, cos⁻¹ | [−1, 1] |
| Range of sin⁻¹ | [−π/2, π/2] |
| Range of cos⁻¹ | [0, π] |
| sin⁻¹x + cos⁻¹x | = π/2 |
| tan⁻¹x + cot⁻¹x | = π/2 |
Common mistakes & exam wins
- Domain of sin⁻¹ and cos⁻¹ is always [−1, 1] — check this before evaluating.
- sin⁻¹x + cos⁻¹x = π/2 is the single most useful identity in this chapter.
- tan⁻¹x has range (−π/2, π/2), an OPEN interval (never reaches the endpoints).
Frequently asked questions
What is the principal value branch of sin⁻¹x?
[−π/2, π/2] — the standard range chosen so sin⁻¹ is a well-defined single-valued function.
What is the domain of cos⁻¹x?
[−1, 1], the same as sin⁻¹x, since both come from the range of sine and cosine.
What is the identity linking sin⁻¹x and cos⁻¹x?
sin⁻¹x + cos⁻¹x = π/2, for all x in [−1, 1].