The principal value of sin⁻¹(sin 10) is: (Note: 10 is in radians)

The principal value of sin⁻¹(sin 10) is: (Note: 10 is in radians)

• A. 10
• B. 3π − 10
• C. 10 − 3π
• D. π − 10

Answer: B) 3π − 10

Explanation: 10 radians ≈ 10×57.3° ≈ 573°. Subtract 2π (≈6.28) → 3.72 rad. Still > π/2 (1.57). sin 10 = sin(10 − 2π) = sin(10 − 6.28) = sin(3.72). Now 3.72 > π/2. sin(3.72) = sin(π − 3.72) = sin(−0.58) no, sin(π − 3.72) = sin(3.14−3.72) = sin(−0.58). Actually sin 3.72 = sin(π − 3.72)? Better: 3.72 − π = 0.58. sin 3.72 = −sin(3.72−π) = −sin 0.58. Principal value must be in [−π/2, π/2]. −sin 0.58 = sin(−0.58). So principal value is −0.58. But 3π − 10 = 9.42−10 = −0.58. Yes, option 1.