Question 17 of 42intermediate🔧 ApplyMCQ1 mark

sin(π/3 - sin⁻¹(-1/2)) is equal to

Exercise: EXERCISE 2.2 | Q: 14 | (Chapter: 13)
For More Understanding

Explanation

To solve this problem, we need to find the principal value of sin⁻¹(-1/2) first, then use it in the given expression. The principal value branch of sin⁻¹x has range [-π/2, π/2]. Since sin(-π/6) = -1/2 and -π/6 lies in this range, the principal value of sin⁻¹(-1/2) is -π/6. Substituting this value and simplifying gives us sin(π/2) = 1.

Solution Steps

  1. Step 1: Find the principal value of sin⁻¹(-1/2). We know sin(-π/6) = -1/2 and -π/6 ∈ [-π/2, π/2]. Therefore, sin⁻¹(-1/2) = -π/6.

  2. Step 2: Substitute in the given expression: sin(π/3 - sin⁻¹(-1/2)) = sin(π/3 - (-π/6)) = sin(π/3 + π/6).

  3. Step 3: Simplify the angle: π/3 + π/6 = 2π/6 + π/6 = 3π/6 = π/2.

  4. Step 4: Calculate sin(π/2) = 1.

  5. Step 5: Therefore, the answer is 1, which corresponds to option (D).

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