Question

$si{n}^{2}0+si{n}^2\frac{\pi }{6}+si{n}^3\frac{\pi }{3}+sin\frac{\pi }{2}$

Answer 1

${\sin }^{2}0+{\sin }^2\frac{\pi }{6}+{\sin }^3\frac{\pi }{3}+\sin \frac{\pi }{2}$

From trigonometric table-

$\sin 0=0$

$\sin \frac{\pi }{6}=\frac{1}{2}$

$\sin \frac{\pi }{3}=\frac{\sqrt{3}}{2}$

$\sin \frac{\pi }{2}=1$

Substituting all these values, we have

${\sin }^{2}0+{\sin }^2\frac{\pi }{6}+{\sin }^3\frac{\pi }{3}+\sin \frac{\pi }{2}$

$=0+{\left(\frac{1}{2}\right)}^2+{\left(\frac{\sqrt{3}}{2}\right)}^3+1$

$=\frac{1}{4}+\frac{3\sqrt{3}}{8}+1$

$=\frac{2+3\sqrt{3}+8}{8}$

$=\frac{10+3\sqrt{3}}{8}$

Hence ${\sin }^{2}0+{\sin }^2\frac{\pi }{6}+{\sin }^3\frac{\pi }{3}+\sin \frac{\pi }{2}=\frac{10+3\sqrt{3}}{8}$

Hence the correct answer is $\frac{10+3\sqrt{3}}{8}$.