Trigonometry & Inverse Trigonometry
Trigonometric Equations with Logarithms
Grade 11
Question:
<p>Number of integral solutions of the equation \(\log_{\sin x} \sqrt{\sin^2 x} + \log_{\cos x} \sqrt{\cos^2 x} = 2\), where \(x \in [0, 6\pi]\)</p>
Step-by-Step Solution
Key Concept: Apply logarithm properties and determine the domain where both bases are positive and not equal to 1.
<p><strong>Step 1:</strong> For the logarithms to be defined, we need $\sin x > 0$, $\sin x \neq 1$, $\cos x > 0$, and $\cos x \neq 1$</p><p><strong>Step 2:</strong> The domain is $x \in \left(0, \frac{\pi}{2}\right) \cup \left(2\pi, \frac{5\pi}{2}\right) \cup \left(4\pi, \frac{9\pi}{2}\right)$</p><p><strong>Step 3:</strong> Number of integral solutions = 1 + 1 + 2 = 4</p>
Correct Answer: 4