Trigonometry & Inverse Trigonometry
Trigonometric Equations
Grade 11
Question:
<p>The sum of all values of \(\theta \in \left[0, \frac{\pi}{2}\right)\) satisfying \(\sin 2\theta + \cos 2\theta = \frac{3}{4}\) is</p>
<p>(a) \(\frac{3\pi}{8}\)</p>
<p>(b) \(\frac{5\pi}{4}\)</p>
<p>(c) \(\frac{\pi}{2}\)</p>
<p>(d) \(\pi\)</p>
Step-by-Step Solution
Key Concept: Express $\sin 2\theta + \cos 2\theta$ as a single sinusoidal function using the identity $a\sin x + b\cos x = \sqrt{a^2+b^2}\sin(x+\phi)$.
<p>The equation $\sin 2\theta + \cos 2\theta = \frac{3}{4}$ can be written as $\sqrt{2}\sin\left(2\theta + \frac{\pi}{4}\right) = \frac{3}{4}$. Solving for $\theta$ in $[0, \frac{\pi}{2})$ and summing the solutions gives $\frac{3\pi}{8}$.</p>
Correct Answer: a