A) \[\ge 0\] Only when \[\theta \ge 0\]
B) \[\le 0\] For all real \[\theta \]
C) \[\ge 0\] For all real \[\theta \]
D) \[\le 0\] Only when \[\theta \le 0\]
Correct Answer: C
Solution :
(c) \[f\left( \theta \right)=\sin \theta \left( \sin \theta +\sin 3\theta \right)\]\[=\left( \sin \theta +3\sin \theta -4{{\sin }^{3}}\theta \right).\sin \theta \]\[=\left( 4\sin \theta -4{{\sin }^{2}}\theta \right)\sin \theta \]\[={{\sin }^{2}}\theta \left( 4-4{{\sin }^{2}}\theta \right)\] \[=4{{\sin }^{2}}\theta \left( 1-{{\sin }^{2}}\theta \right)\]\[=4{{\sin }^{2}}\theta {{\cos }^{2}}\theta ={{\left( 2\sin \theta \cos \theta \right)}^{2}}\] \[={{\left( \sin 2\theta \right)}^{2}}\ge 0\] Which is true for all\[\theta \].
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