If and , then is equal to
- A.0
- B.10
- C.
- D.
Show answer and solution
Correct option: D
Solution
sin⁻¹(sin θ) maps to [−π/2, π/2] and cos⁻¹(cos θ) maps to [0, π].
Since 10 radians lies in [3π, 7π/2] (as 3π ≈ 9.42 and 7π/2 ≈ 11.0):
x = sin⁻¹(sin 10) = 3π − 10 (reflecting 10 into [−π/2, π/2]: since 10 is in (3π, 7π/2), sin 10 = sin(3π − 10) and 3π − 10 ∈ (−π/2, 0) ✓)
y = cos⁻¹(cos 10) = 4π − 10 (since cos(4π − 10) = cos 10 and 4π − 10 ∈ (0, π) ✓)
y − x = (4π − 10) − (3π − 10) = π.
Since 10 radians lies in [3π, 7π/2] (as 3π ≈ 9.42 and 7π/2 ≈ 11.0):
x = sin⁻¹(sin 10) = 3π − 10 (reflecting 10 into [−π/2, π/2]: since 10 is in (3π, 7π/2), sin 10 = sin(3π − 10) and 3π − 10 ∈ (−π/2, 0) ✓)
y = cos⁻¹(cos 10) = 4π − 10 (since cos(4π − 10) = cos 10 and 4π − 10 ∈ (0, π) ✓)
y − x = (4π − 10) − (3π − 10) = π.
[JEE Main 2019 · 9 Jan · Shift II]