Let f( ) = ( + 3 ), then f( ) - Vidyadip

MCQ

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

A
$ \ge 0$ only when $\theta \ge 0$
B
$ \le 0$ for all real $\theta $
✓
$ \ge 0$ for all real $\theta $
D
$ \le 0$ only when $\theta \le 0$

✓

Answer

Correct option: C.

$ \ge 0$ for all real $\theta $

c
(c) Here, $f(\theta ) = \sin \theta (\sin \theta + \sin 3\theta )$

$ = \sin \theta (\sin \theta + 3\sin \theta - 4{\sin ^3}\theta ) = 4{\sin ^2}\theta (1 - {\sin ^2}\theta )$

$ = 4{\sin ^2}\theta {\cos ^2}\theta = {(\sin 2\theta )^2}$

$\therefore$ $f(\theta ) \ge 0$ for all real $\theta $.

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