Question 24 of 42intermediate🔧 ApplyShort Answer2 marks

Find the values of each of the expressions: tan⁻¹(tan(3π/4))

Correct Answer

-π/4

Exercise: EXERCISE 2.2 | Q: 11 | (Page 13)
For More Understanding

Explanation

The question is from Exercise 11. The principal value branch of tan⁻¹ is (-π/2, π/2). Since 3π/4 is not in this interval, we cannot directly apply tan⁻¹(tan x) = x. Instead, we compute tan(3π/4) = -1 and find the angle in (-π/2, π/2) with tangent -1, which is -π/4. This is analogous to Example 6 where sin⁻¹(sin x) is adjusted to the principal branch.

Solution Steps

  1. Step 1: Evaluate tan(3π/4) = tan(π - π/4) = -tan(π/4) = -1.

  2. Step 2: Recall that the principal value branch of tan⁻¹ is (-π/2, π/2).

  3. Step 3: Find angle θ ∈ (-π/2, π/2) such that tan θ = -1. This angle is θ = -π/4.

  4. Step 4: Therefore, tan⁻¹(tan(3π/4)) = -π/4.

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