Functions 1 Question 8
8. Let $f(\theta)=\sin \theta(\sin \theta+\sin 3 \theta)$. Then, $f(\theta)$
(2000, 1M)
(a) $\geq 0$, only when $\theta \geq 0$
(b) $\leq 0$, for all real $\theta$
(c) $\geq 0$, for all real $\theta$
(d) $\leq 0$, only when $\theta \leq 0$
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Answer:
Correct Answer: 8. $A \rightarrow$
Solution:
- It is given,
$$ \begin{aligned} f(\theta) & =\sin \theta(\sin \theta+\sin 3 \theta) \\ & =\left(\sin \theta+3 \sin \theta-4 \sin ^{3} \theta\right) \sin \theta \\ & =\left(4 \sin \theta-4 \sin ^{3} \theta\right) \sin \theta=\sin ^{2} \theta\left(4-4 \sin ^{2} \theta\right) \\ & =4 \sin ^{2} \theta \cos ^{2} \theta=(2 \sin \theta \cos \theta)^{2} \\ & =(\sin 2 \theta)^{2} \geq 0 \end{aligned} $$
which is true for all $\theta$.
