Matrices & Determinants
Determinant Properties and Trigonometric Equations
Grade 12
Question:
<p>If $\begin{vmatrix} 1+\cos 2\theta & \sin 2\theta & 2\sqrt{3}\tan\theta \\ \cos 2\theta & 1+\sin 2\theta & 2\sqrt{3}\tan\theta \\ \cos 2\theta & \sin 2\theta & 1+2\sqrt{3}\tan\theta \end{vmatrix} = 0$, then $\theta$ may be:</p>
<p>(a) $\frac{\pi}{6}$</p>
<p>(b) $\frac{5\pi}{6}$</p>
<p>(c) $\frac{7\pi}{6}$</p>
<p>(d) $\frac{11\pi}{6}$</p>
Step-by-Step Solution
Key Concept: Use row and column operations to simplify the determinant, then apply trigonometric identities.
<p>Perform row operations on the determinant to simplify it. Apply column operations and use trigonometric identities to reduce to a form that equals zero.</p>
Correct Answer: a, b, c, d