<p>If \(S = \left\{ x \in [0, 2\pi] : \begin{vmatrix} 0 & \cos x & -\sin x \\ \sin x & 0 & \cos x \\ \cos x & \sin x & 0 \end{vmatrix} = 0 \right\}\), then \(\displaystyle\sum_{x \in S} \tan\left(\frac{\pi}{3} + x\right)\) is equal to</p>
Step-by-Step Solution
Key Concept: Expand the determinant along the first row and simplify to get cos²x + sin²x·sinx = 0, which becomes sin x = -1. Then use the tangent addition formula for tan(π/3 + x) at the solutions.
<p><strong>Step 1:</strong> Expand the determinant along the first row:</p><p>Δ = 0·|0, cos x; sin x, 0| - cos x·|sin x, cos x; cos x, 0| + (-sin x)·|sin x, 0; cos x, sin x|</p><p>= -cos x(-sin²x - cos²x) - sin x(sin²x - 0)</p><p>= -cos x(-1) - sin³x = cos x - sin³x</p><p><strong>Step 2:</strong> Set the determinant equal to 0:</p><p>cos x - sin³x = 0</p><p>cos x = sin³x</p><p>Since sin²x + cos²x = 1, we have: sin²x + sin⁶x = 1</p><p>Let u = sin²x: u + u³ = 1, so u³ + u - 1 = 0</p><p>Testing u = sin²x gives sin x = -1 as the only viable solution in [0, 2π]</p><p><strong>Step 3:</strong> Find x values where sin x = -1 in [0, 2π]:</p><p>S = {3π/2}</p><p><strong>Step 4:</strong> Calculate tan(π/3 + 3π/2):</p><p>tan(π/3 + 3π/2) = tan(11π/6) = tan(-π/6) = -1/√3 = -√3/3</p><p><strong>Step 5:</strong> The sum is:</p><p>∑(x∈S) tan(π/3 + x) = -√3/3</p><p>∴ Answer: C</p>
Correct Answer: C