Question

If $f\left(\theta \right)=si{n}^2\theta +si{n}^2(\theta +\frac{2\pi }{3})+si{n}^2(\theta +\frac{4\pi }{3}),thenf\left(\frac{\pi }{15}\right)$

• A. $\frac{2}{3}$
• B. $\frac{3}{2}$
• C. $\frac{1}{2}$
• D. $\frac{1}{3}$

Answer 1

The correct option is B

$\frac{3}{2}$

$f\left(\theta \right)=si{n}^2\theta +si{n}^2(\theta +\frac{2\pi }{3})+si{n}^2(\theta +\frac{4\pi }{3})=\frac{1-cos2\theta }{2}+\frac{1-cos(2\theta +\frac{4\pi }{3})}{2}+\frac{1-cos(2\theta +8\frac{\pi }{3})}{2}=\frac{1}{2}(3-cos2\theta -cos(\frac{4\pi }{3}+2\theta )-cos(\frac{8\pi }{3}+2\theta \left)\right)=\frac{1}{2}(3-cos2\theta -2cos(2\pi +2\theta \left)cos\frac{2\pi }{3}\right)=\frac{1}{2}(3-cos2\theta +cos2\theta )f\left(\theta \right)=\frac{3}{2}\therefore f\left(\frac{\pi }{15}\right)=\frac{3}{2}$