Matrices & Determinants
Trigonometric determinants
Grade 12
Question:
<p>If \(\sin^2\theta (1 + \sin^2\theta)\) <span class="MathJax">\(\cos^2\theta\)</span> <span class="MathJax">\(4\sin^4\theta\)</span> <span class="MathJax">\(\sin^2\theta\)</span> <span class="MathJax">\(1 + \cos^2\theta\)</span> <span class="MathJax">\(4\sin^4\theta\)</span> <span class="MathJax">\(\sin\theta\)</span> <span class="MathJax">\(\cos\theta\)</span> <span class="MathJax">\(1 + 4\sin^4\theta\)</span> = 0, then \(\theta\) is equal to</p>
<p>(a) \(\frac{7\pi}{24}, \frac{11\pi}{24}\)</p>
<p>(b) \(\frac{5\pi}{24}, \frac{7\pi}{24}\)</p>
<p>(c) \(\frac{11\pi}{24}, \frac{\pi}{24}\)</p>
<p>(d) \(\frac{\pi}{24}, \frac{7\pi}{24}\)</p>
Step-by-Step Solution
Key Concept: Expand the determinant, simplify using trigonometric identities, and solve the resulting equation.
<p>Expand the determinant and set it equal to zero. This yields a trigonometric equation in \(\sin\theta\) and \(\cos\theta\). Solve to find \(\theta = \frac{7\pi}{24}\) and \(\frac{11\pi}{24}\).</p>
Correct Answer: A