Let f(θ ) = sin θ (sin θ + sin 3θ ) , then f(θ )

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1.Relation and Function

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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

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