# Let $f(\theta)=\sin \theta(\sin \theta+\sin 3\theta)$.Then $f(\theta)$ is

$\begin{array}{1 1}a)\;\geq 0\;only\;when\;\theta\geq 0\\b)\;\leq 0\;for \;all\;real\;\theta\\c)\;\geq 0\;for\;all\;real\;\theta\\d)\;\leq 0\;only\;when\;\theta\leq 0\end {array}$

$F(\theta)=\sin\theta(\sin\theta+\sin 3\theta)$
$\quad\;\;\;\;=(\sin\theta+3\sin\theta-4\sin^3\theta).\sin\theta$
$\quad\;\;\;\;=(4\sin\theta-4\sin^3\theta)\sin\theta$
$\quad\;\;\;\;=\sin^2\theta(4-4\sin^2\theta)$
$\quad\;\;\;\;=4\sin^2\theta(1-\sin^2\theta)$
$1-\sin^2\theta=\cos^2\theta$
$\quad\;\;\;\;=4\sin^2\theta(\cos^2\theta)$
$\quad\;\;\;\;=(2\sin\theta\cos\theta)^2$
$2\sin\theta\cos\theta=\sin 2\theta$
$\quad\;\;\;\;=(\sin 2\theta)^2 \geq 0$
Which is true for all $\theta$