<p>The principal value of \(\tan^{-1}(-\sqrt{3})\) is ______.</p>
Step-by-Step Solution
Key Concept: The principal value of inverse tangent is defined on the range (-π/2, π/2). Since tan(-π/3) = -√3 and -π/3 lies in this range, tan⁻¹(-√3) = -π/3.
<p><strong>Step 1:</strong> Recall that tan⁻¹(x) has principal value range: (-π/2, π/2)</p><p><strong>Step 2:</strong> We need to find θ ∈ (-π/2, π/2) such that tan(θ) = -√3</p><p><strong>Step 3:</strong> We know tan(π/3) = √3, so tan(-π/3) = -√3</p><p><strong>Step 4:</strong> Since -π/3 ∈ (-π/2, π/2), this is the principal value</p><p><strong>Verification:</strong> tan(-π/3) = -tan(π/3) = -√3 ✓</p><p>∴ Answer: <strong>-π/3</strong></p>
Correct Answer: -π/3