Principal values of inverse trigonometric functions: Class 12 practice
Principal value ranges: sin⁻¹ in [−π/2, π/2], cos⁻¹ in [0, π], tan⁻¹ in (−π/2, π/2). Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Standard trigonometric values
The idea behind principal values of inverse trigonometric functions
Principal value ranges: sin⁻¹ in [−π/2, π/2], cos⁻¹ in [0, π], tan⁻¹ in (−π/2, π/2). Choose the angle in that range whose ratio matches.
Step-by-step method
- Find an angle with the given ratio.
- Adjust it into the principal range.
- Give the answer in radians.
Worked example 1 (Basic)
Find the principal value of cos⁻¹(−1/2).
- Principal range checked
- = 2π/3
- Answer: 2π/3
Worked example 2 (Standard)
Find the principal value of tan⁻¹(−√3).
- Principal range checked
- = −π/3
- Answer: −π/3
Worked example 3 (Challenge)
Find the principal value of sin⁻¹(1/2).
- Principal range checked
- = π/6
- Answer: π/6
Common mistakes and misconceptions
- Giving cos⁻¹(−1/2) as −π/3
- Answering outside the principal range
Practice questions
- (Basic) Find the principal value of tan⁻¹(1/√3).
- (Basic) Find the principal value of tan⁻¹(1).
- (Standard) Find the principal value of sin⁻¹(√3/2).
- (Standard) Find the principal value of cos⁻¹(1/2).
- (Challenge) Find the principal value of cos⁻¹(−1/√2).
- (Challenge) Find the principal value of cos⁻¹(−√3/2).
Answer key with full working
- Q1: Principal range checked → = π/6 → Answer: π/6
- Q2: Principal range checked → = π/4 → Answer: π/4
- Q3: Principal range checked → = π/3 → Answer: π/3
- Q4: Principal range checked → = π/3 → Answer: π/3
- Q5: Principal range checked → = 3π/4 → Answer: 3π/4
- Q6: Principal range checked → = 5π/6 → Answer: 5π/6
Syllabus note
This skill is listed under "Inverse Trigonometric Functions" for Class 12 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is principal values of inverse trigonometric functions in?
"Inverse Trigonometric Functions", Class 12 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.