Matrices & Determinants
Trigonometric determinants
Grade 12
Question:
<p>When the determinant \(\begin{vmatrix} \cos 2x & \sin^2 x & \cos 4x \\ \sin^2 x & \cos 2x & \cos^2 x \\ \cos 4x & \cos^2 x & \cos 2x \end{vmatrix}\) is expanded in powers of \(\sin x\), then the constant term in that expression is</p>
<p>1</p>
<p>0</p>
<p>−1</p>
<p>2</p>
Step-by-Step Solution
Key Concept: Use trigonometric identities (cos 2x = 1 - 2sin²x, cos 4x = 1 - 8sin²x + 8sin⁴x) to express the determinant as a polynomial in sin x, then identify the constant term by setting sin x = 0.
<p><strong>Step 1:</strong> Express all terms using sin x. Let s = sin x, then:</p><ul><li>cos 2x = 1 - 2s²</li><li>sin²x = s²</li><li>cos 4x = 1 - 8s² + 8s⁴</li><li>cos²x = 1 - s²</li></ul><p><strong>Step 2:</strong> The determinant becomes:</p><p>$\begin{vmatrix} 1-2s² & s² & 1-8s²+8s⁴ \\ s² & 1-2s² & 1-s² \\ 1-8s²+8s⁴ & 1-s² & 1-2s² \end{vmatrix}$</p><p><strong>Step 3:</strong> The constant term (terms with no s) is found by setting s = 0:</p><p>$\begin{vmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 1 & 1 & 1 \end{vmatrix}$</p><p><strong>Step 4:</strong> Expand along row 1: = 1(1-1) - 0 + 1(0-1) = 0 - 1 = -1</p><p>∴ Answer: <strong>C</strong> (constant term = -1)</p>
Correct Answer: C