Inverse Trigonometric Functions Class 12 Notes

• If
• $x\in [-1,1]$
• ,
• ${\sin }^{-1}(\sin y)=y$
• only when
• $y\in [-\pi /2,\pi /2]$
• , not outside the branch.

• Convert the equation into a single inverse function, then apply the appropriate identity.
• Check every candidate solution against the principal value range.
• Use substitutions like
• ${\tan }^{-1}x=\theta$
• to turn an inverse-trig equation into a plain trig equation.

Example: Find the principal value of ${\sin }^{-1}\left(-\frac{1}{2}\right)$.

Solution: On the principal branch $[-\pi /2,\pi /2]$, $\sin (-\pi /6)=-1/2$. Using the negative identity, ${\sin }^{-1}(-1/2)=-{\sin }^{-1}(1/2)=-\pi /6$.

Example: Evaluate ${\tan }^{-1}(1/2)+{\tan }^{-1}(1/3)$.

Solution: With $xy=1/6<1$, the sum identity gives ${\tan }^{-1}\frac{1/2+1/3}{1-1/6}={\tan }^{-1}\frac{5/6}{5/6}={\tan }^{-1}1=\pi /4$.

Exercises (NCERT Q1–Q35)

• 1) Find the principal value of .
• 2) Find the principal value of .
• 3) Find the principal value of .
• 4) Find the principal value of .
• 5) Find the principal value of .
• 6) Find the principal value of .
• 7) Find the principal value of .
• 8) Find the principal value of .
• 9) Find the principal value of .
• 10) Find the principal value of .
• 11) Find the value of .
• 12) Find the value of .
• 13) If , then (A) ; (B) ; (C) ; (D) .
• 14) is equal to (A) ; (B) ; (C) ; (D) .
• 15) Prove that , .
• 16) Prove that , .
• 17) Prove that .
• 18) Prove that .
• 19) Write , , in the simplest form.
• 20) Write , , in the simplest form.
• 21) Write , , in the simplest form.
• 22) Write , , in the simplest form.
• 23) Write , , in the simplest form.
• 24) Write , , , in the simplest form.
• 25) Find the value of .
• 26) Find the value of .
• 27) Find the value of , , , .
• 28) If , then find the value of .
• 29) If , then find the value of .
• 30) Find the value of .
• 31) Find the value of .
• 32) Find the value of .
• 33) is equal to (A) ; (B) ; (C) ; (D) .
• 34) is equal to (A) ; (B) ; (C) ; (D) 1.
• 35) is equal to (A) ; (B) ; (C) 0; (D) .

Answers at a glance

1. 1
   Q1 → answer not in source — verify against the textbook
2. 2
   Q2 → answer not in source — verify against the textbook
3. 3
   Q3 → answer not in source — verify against the textbook
4. 4
   Q4 → answer not in source — verify against the textbook
5. 5
   Q5 → answer not in source — verify against the textbook
6. 6
   Q6 → answer not in source — verify against the textbook
7. 7
   Q7 → answer not in source — verify against the textbook
8. 8
   Q8 → answer not in source — verify against the textbook
9. 9
   Q9 → answer not in source — verify against the textbook
10. 10
    Q10 → answer not in source — verify against the textbook
11. 11
    Q11 → answer not in source — verify against the textbook
12. 12
    Q12 → answer not in source — verify against the textbook
13. 13
    Q13 → Option (B) —
14. 14
    Q14 → Option (B) —
15. 15
    Q15 → answer not in source — verify against the textbook
16. 16
    Q16 → answer not in source — verify against the textbook
17. 17
    Q17 → answer not in source — verify against the textbook
18. 18
    Q18 → answer not in source — verify against the textbook
19. 19
    Q19 → answer not in source — verify against the textbook
20. 20
    Q20 → For x > 1: \text{cosec}^{-1}x; for x < −1: -\text{cosec}^{-1}x
21. 21
    Q21 → answer not in source — verify against the textbook
22. 22
    Q22 → π/4 − x for 0 < x < 3π/4; 5π/4 − x for 3π/4 < x < π.
23. 23
    Q23 → answer not in source — verify against the textbook
24. 24
    Q24 → answer not in source — verify against the textbook
25. 25
    Q25 → answer not in source — verify against the textbook
26. 26
    Q26 → answer not in source — verify against the textbook
27. 27
    Q27 → answer not in source — verify against the textbook
28. 28
    Q28 → answer not in source — verify against the textbook
29. 29
    Q29 → answer not in source — verify against the textbook
30. 30
    Q30 → answer not in source — verify against the textbook
31. 31
    Q31 → answer not in source — verify against the textbook
32. 32
    Q32 → answer not in source — verify against the textbook
33. 33
    Q33 → Option (B) —
34. 34
    Q34 → Option (D) — 1.
35. 35
    Q35 → Option (B) —

MCQ Questions

1 Mark Questions

1. 1
   Find the principal value of cos⁻¹(1/2). (Ans: π/3)
2. 2
   Write the domain of sin⁻¹x. (Ans: [−1, 1])
3. 3
   Find sin⁻¹(sin(3π/4)). (Ans: π/4)
4. 4
   State the range of sec⁻¹x. (Ans: [0, π] − {π/2})

2 Mark Questions

1. 1
   Find the value of tan⁻¹(1) + cos⁻¹(−1/2) + sin⁻¹(−1/2). (Ans: principal values give π/4 + 2π/3 − π/6 = 3π/4)
2. 2
   Evaluate cos(tan⁻¹(3/4)). (Ans: 4/5)
3. 3
   Simplify 2 tan⁻¹(1/3) + tan⁻¹(1/7). (Ans: π/4)

3 Mark Questions

1. 1
   Show that tan⁻¹(1/2) + tan⁻¹(1/3) = π/4 using the formula tan⁻¹x + tan⁻¹y = tan⁻¹[(x + y)/(1 − xy)].
2. 2
   Prove 2 tan⁻¹x = sin⁻¹[2x/(1 + x²)] for x ∈ (−1, 1).
3. 3
   Solve tan⁻¹[(1 − x)/(1 + x)] = (1/2) tan⁻¹x, x > 0. (Ans: x = 1/√3)

4 Mark Questions

1. 1
   Prove tan⁻¹(1) + tan⁻¹(2) + tan⁻¹(3) = π, then generalise: for x ≥ 0, tan⁻¹x + tan⁻¹(1/x) = π/2. State the conditions under which tan⁻¹x + tan⁻¹y = tan⁻¹[(x + y)/(1 − xy)].
2. 2
   Sketch the meaning of a principal value branch for sin⁻¹x and cos⁻¹x, state their domains, ranges and graphs' end behaviour, and evaluate sin(cos⁻¹(4/5)) + cos(sin⁻¹(12/13)). (Ans: 3/5 + 12/13 = 99/65)