Question 23 of 42intermediate🔧 ApplyShort Answer2 marks

Find the values of each of the expressions: sin⁻¹(sin(2π/3))

Correct Answer

π/3

Exercise: EXERCISE 2.2 | Q: 10 | (Page 13)
For More Understanding

Explanation

This question tests the understanding of the principal branch of inverse trigonometric functions. The principal branch of sin⁻¹x is [-π/2, π/2]. Since 2π/3 lies outside this interval, we cannot directly apply sin⁻¹(sin x) = x. The student must use the property sin(π - θ) = sin θ to convert the angle to one within the principal branch.

Solution Steps

  1. Step 1: We know that sin⁻¹(sin x) = x only when x ∈ [-π/2, π/2], which is the principal branch of sin⁻¹x.

  2. Step 2: Here, 2π/3 ∉ [-π/2, π/2], so we cannot directly write sin⁻¹(sin(2π/3)) = 2π/3.

  3. Step 3: Using the property sin(π - θ) = sin θ, we have sin(2π/3) = sin(π - 2π/3) = sin(π/3).

  4. Step 4: Since π/3 ∈ [-π/2, π/2], we get sin⁻¹(sin(2π/3)) = sin⁻¹(sin(π/3)) = π/3.

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