Question 21 of 42intermediate🔧 ApplyShort Answer2 marks

Find the values of the following: tan⁻¹[2cos(2sin⁻¹(1/2))]

Correct Answer

π4\frac{\pi}{4}

Exercise: EXERCISE 2.2 | Q: 8 | (Page 12)
For More Understanding

Explanation

This question tests the understanding of inverse trigonometric functions and their principal values. Students need to evaluate the expression from inside out, starting with sin⁻¹(1/2), then applying the double angle, cosine, and finally tan⁻¹. The key is knowing the standard values: sin(π/6) = 1/2, cos(π/3) = 1/2, and tan(π/4) = 1.

Solution Steps

  1. Step 1: Find sin⁻¹(1/2). Since sin(π/6) = 1/2, we have sin⁻¹(1/2) = π/6.

  2. Step 2: Calculate 2sin⁻¹(1/2) = 2 × π/6 = π/3.

  3. Step 3: Find cos(π/3) = 1/2.

  4. Step 4: Calculate 2cos(2sin⁻¹(1/2)) = 2 × (1/2) = 1.

  5. Step 5: Find tan⁻¹(1). Since tan(π/4) = 1, we have tan⁻¹(1) = π/4.

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