Question

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

• A. $\ge 0onlywhen\theta \ge 0$
• B. $\le 0forallral\theta$
• C. $\ge 0forallreal\theta$
• D. $\le 0onlywhen\theta \le 0$

Answer 1

The correct option is C $\ge 0forallreal\theta$

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