Find the principal values of the following: tan⁻¹(-1)
Let tan⁻¹(-1) = y. Then, tan y = -1. We know that tan(π/4) = 1. Since tangent is an odd function, tan(-π/4) = -tan(π/4) = -1.
The principal value branch of tan⁻¹ has range (-π/2, π/2). The value -π/4 lies within this interval.
Hence, the principal value of tan⁻¹(-1) is -π/4.
Explanation
This question follows the same pattern as Example 2 in the textbook. The key steps are: (1) Let the inverse function equal y, (2) Express the given value in terms of a known angle, (3) Verify the result lies within the principal value branch range. For tan⁻¹, the principal value branch is (-π/2, π/2), so negative angles within this range are valid principal values.
Solution Steps
Step 1: Let tan⁻¹(-1) = y, then tan y = -1
Step 2: Express -1 as tan(-π/4) since tan(π/4) = 1 and tan is odd
Step 3: Verify -π/4 lies in the principal value branch (-π/2, π/2)
Step 4: Conclude principal value is -π/4