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1.Relation and Function
medium
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 $
C
$ \ge 0$ for all real $\theta $
D
$ \le 0$ only when $\theta \le 0$
(IIT-2000)
Solution
(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 $.
Standard 12
Mathematics
