Question

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

• A. $\ge 0onlywhen\theta \ge 0$
• B. $\le 0forallral\theta$
• C. $\ge 0forallreal\theta$
• D. $\le 0onlywhen\theta \le 0$

Answer 1

The correct option is C $\ge 0forallreal\theta$

$f\left(\theta \right)=sin\theta (sin\theta +sin3\theta )=(sin\theta +3sin\theta -4si{n}^3\theta ).sin\theta =(4sin\theta -4si{n}^3\theta )sin\theta =si{n}^2\theta (4-4si{n}^2\theta )=4si{n}^2\theta (1-si{n}^2\theta )=4si{n}^2\theta co{s}^2\theta =(2sin\theta cos\theta {)}^2=(sin2\theta {)}^2\ge 0$\\
which is true for all $\theta .$