Question

Let $f\left(\theta \right)=sin\theta (sin\theta +sin3\theta )$, then $f\left(\theta \right)$

• A. ≥ 0 only when θ≥ 0
• B. ≤ 0 for all real θ
• C. ≥ 0 for all real θ
• D. ≤ 0 only when θ ≤0

Answer 1

The correct option is C ≥ 0 for all real θ

Here, $f\left(\theta \right)=sin\theta (sin\theta +sin3\theta )$

$=sin\theta (sin\theta +3sin\theta -4si{n}^3\theta )=4si{n}^2\theta (1-si{n}^2\theta )$

$=4si{n}^2\theta co{s}^2\theta =(sin2\theta {)}^2$

$\therefore f\left(\theta \right)\ge 0$ for all real $\theta .$