Question 30 of 42intermediate🔧 ApplyShort Answer2 marks

Find the value of the following: tan⁻¹(tan(7π/6))

Correct Answer

To find tan⁻¹(tan(7π/6)), we first note that the principal branch of tan⁻¹x is (-π/2, π/2).

Since 7π/6 = π + π/6, which is greater than π/2, this angle lies outside the principal branch. Therefore, tan⁻¹(tan x) ≠\neq x directly.

Using the periodicity of tangent: tan(7π/6) = tan(π + π/6) = tan(π/6). Since π/6 lies within (-π/2, π/2), we get tan⁻¹(tan(7π/6)) = tan⁻¹(tan(π/6)) = π/6.

Exercise: Miscellaneous Exercise on Chapter 2 | Q: 2 | (Page 14)
For More Understanding

Explanation

This question tests understanding of inverse trigonometric functions and their principal branches. The key insight is that tan⁻¹(tan x) = x only when x lies in the principal branch (-π/2, π/2). For angles outside this range, we must find an equivalent angle within the range using the periodicity of tangent (period = π). The textbook context shows similar examples in the Miscellaneous Exercise.

Solution Steps

  1. Step 1: Identify principal branch of tan⁻¹x as (-π/2, π/2).

  2. Step 2: Check that 7π/6 > π/2, so it's outside the principal branch.

  3. Step 3: Apply periodicity: tan(7π/6) = tan(π + π/6) = tan(π/6).

  4. Step 4: Since π/6 ∈ (-π/2, π/2), the answer is π/6.

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