Question

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

• 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

The correct option is C $\ge 0$ for all real $\theta$

$f\left(\theta \right)=\frac{1}{2}[(1-\cos 2\theta )+(\cos 2\theta -\cos 4\theta )]$

$[\because 1-\cos 2\theta =2{\sin }^2\theta$and $2\sin asinb=cos(a-b)-cos(a+b)$]

$=\frac{1}{2}(1-\cos 4\theta )={\sin }^{2}2\theta \ge 0\forall$ real $\theta$.