Question

Let $f\left(\theta \right)=\sin \theta (\sin \theta +\sin 3\theta )$, then $f\left(\theta \right)$ is

• A. $\ge 0\text{only when}\theta \ge 0$
• B. $\le 0\text{for all real}\theta$
• C. $\ge 0\text{for all real}\theta$
• D. $\le 0\text{only when}\theta \le 0$

Answer 1

The correct option is C $\ge 0\text{for all real}\theta$

$f\left(\theta \right)=\sin \theta (\sin \theta +\sin 3\theta )=(\sin \theta +3\sin \theta -4{\sin }^3\theta )\sin \theta =(4\sin \theta -4{\sin }^3\theta )\sin \theta =4{\sin }^2\theta (1-{\sin }^2\theta )=4{\sin }^2\theta {\cos }^2\theta \ge 0$
Which is true for all $\theta \in R$