Question

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

Answer: (C)

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

$f\left(\theta \right)=\sin \theta (\sin \theta +3\sin \theta -4{\sin }^3\theta )$

$f\left(\theta \right)=4{\sin }^2\theta (1-{\sin }^2\theta )$

$f\left(\theta \right)=4{\sin }^2\theta {\cos }^2\theta$

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

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