Question

Let $f\left(\theta \right)=sin\theta .(sin\theta +sin3\theta )$, then $f\left(\theta \right)$ is

• 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 $\theta \le 0$

Answer 1

Answer: (C)

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

$f\left(\theta \right)=\sin \theta (\sin \theta +\sin 3\theta )$

$f\left(\theta \right)={\sin }^2\theta +\sin \theta \sin 3\theta$

$f\left(\theta \right)=\frac{1}{2}[2{\sin }^2\theta +2\sin \theta \sin 3\theta ]$ $(\because$ $1-cos2x=2si{n}^{2}x,2\sin c\sin d=\cos (c+d)-\cos (c-d))$

$f\left(\theta \right)=\frac{1}{2}(1-cos2\theta )+\frac{1}{2}(cos2\theta -cos4\theta )$

$=\frac{1}{2}(1-cos4\theta )$

$\Rightarrow f\left(\theta \right)=si{n}^{2}2\theta \ge 0$

$\Rightarrow f\left(\theta \right)\ge 0$ for all real $\theta$.