Example 1 · level 1
Find the principal value of tan⁻¹(√3).
- 2π/3
- −π/3
- 4π/3
- π/3
Show solution
- The principal branch of tan⁻¹ is (−π/2, π/2).
- tan(π/3) = √3 and π/3 lies in that branch.
- So tan⁻¹(√3) = π/3.
Answer: π/3
CLASS 12 MATHS · INVERSE TRIGONOMETRIC FUNCTIONS
Principal values of inverse trigonometric functions
The principal value must lie in the principal branch: sin⁻¹ in [−π/2, π/2], cos⁻¹ in [0, π], tan⁻¹ in (−π/2, π/2), cot⁻¹ in (0, π).
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Worked examples
Example 2 · level 2
Find the principal value of sin⁻¹(−1/2).
- −π/6
- 5π/6
- π/6
- 7π/6
Show solution
- The principal branch of sin⁻¹ is [−π/2, π/2].
- sin(−π/6) = −1/2 and −π/6 lies in that branch.
- So sin⁻¹(−1/2) = −π/6.
Answer: −π/6
Example 3 · level 3
Find the value of tan⁻¹(−1/√3) − cot⁻¹(−√3).
- π
- −π
- 2π
- 0
Show solution
- tan⁻¹(−1/√3) = −π/6, since tan(−π/6) = −1/√3 and −π/6 lies in (−π/2, π/2).
- cot⁻¹(−√3) = 5π/6, since cot(5π/6) = −√3 and 5π/6 lies in (0, π).
- −π/6 − 5π/6 = −π.
Answer: −π