Trigonometry & Inverse Trigonometry
Product-to-Sum Identity — Sum of θ where cos 3θ is Maximum
nta_pyq_2024_apr
Grade 11
Question:
Let $|\cos\theta\cos(60^\circ-\theta)\cos(60^\circ+\theta)|\leq\dfrac{1}{8}$, $\theta\in[0,2\pi]$. Then, the sum of all $\theta\in[0,2\pi]$, where $\cos3\theta$ attains its maximum value, is:
$15\pi$
$18\pi$
$6\pi$
$9\pi$
Step-by-Step Solution
Key Concept: Identity: $\cos\theta\cdot\cos(60^\circ-\theta)\cdot\cos(60^\circ+\theta)=\dfrac{1}{4}\cos3\theta$. So $\left|\dfrac{1}{4}\cos3\theta\right|\leq\dfrac{1}{8}\Rightarrow|\cos3\theta|\leq\dfrac{1}{2}$.
$\theta=\pi/9,5\pi/9,7\pi/9,11\pi/9,13\pi/9,17\pi/9$. Sum $=6\pi$.
Correct Answer: 3