Question 11 of 42intermediate🔧 ApplyShort Answer2 marks

Find the values of the following: tan⁻¹(1) + cos⁻¹(-1/2) + sin⁻¹(-1/2)

Correct Answer

3π/4

Exercise: EXERCISE 2.1 | Q: 11 | (Page 10)
For More Understanding

Explanation

This question requires finding the principal values of each inverse trigonometric function and then adding them. The principal value ranges are: tan⁻¹ ∈ (-π/2, π/2), cos⁻¹ ∈ [0, π], and sin⁻¹ ∈ [-π/2, π/2]. Each value is computed within its respective principal branch.

Solution Steps

  1. Step 1: Find tan⁻¹(1). Since tan(π/4) = 1 and π/4 lies in the principal range (-π/2, π/2), we have tan⁻¹(1) = π/4

  2. Step 2: Find cos⁻¹(-1/2). Since cos(2π/3) = -1/2 and 2π/3 lies in the principal range [0, π], we have cos⁻¹(-1/2) = 2π/3

  3. Step 3: Find sin⁻¹(-1/2). Since sin(-π/6) = -1/2 and -π/6 lies in the principal range [-π/2, π/2], we have sin⁻¹(-1/2) = -π/6

  4. Step 4: Add all values: π/4 + 2π/3 - π/6 = 3π/12 + 8π/12 - 2π/12 = 9π/12 = 3π/4

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