Question
Let $$f\left( \theta \right) = \sin \theta \left( {\sin \theta + \sin 3\theta } \right).$$ Then $$f\left( \theta \right)$$ is
A.
$$ \geqslant 0\,\,{\text{only when }}\theta \geqslant {\text{0}}$$
B.
$$ \leqslant 0\,{\text{for all real}}\,\theta $$
C.
$$ \geqslant 0\,{\text{for all real}}\,\theta $$
D.
$$ \leqslant 0\,\,{\text{only when }}\theta \leqslant {\text{0}}$$
Answer :
$$ \geqslant 0\,{\text{for all real}}\,\theta $$
Solution :
$$\eqalign{
& f\left( \theta \right) = \sin \theta \left( {\sin \theta + \sin 3\theta } \right) \cr
& = \left( {\sin \theta + 3\sin \theta - 4{{\sin }^3}\theta } \right).\sin \theta \cr
& = \left( {4\sin \theta - 4{{\sin }^3}\theta } \right)\sin \theta = {\sin ^2}\theta \left( {4 - 4{{\sin }^2}\theta } \right) \cr
& = 4{\sin ^2}\theta \left( {1 - {{\sin }^2}\theta } \right) \cr
& = 4{\sin ^2}\theta {\cos ^2}\theta \cr
& = {\left( {2\sin \theta \cos \theta } \right)^2} \cr
& = {\left( {\sin 2\theta } \right)^2} \geqslant 0 \cr} $$
which is true for all $$\theta .$$