Question

Find: $\sin 20\times \sin 40\times \sin 60\times \sin 80$.

• A. $\frac{3}{60}$
• B. $\frac{3}{16}$
• C. $\frac{3}{20}$
• D. $\frac{3}{40}$

Answer 1

The correct option is B $\frac{3}{16}$

$\sin \theta \sin (60-\theta )\sin (60+\theta )$

$=\sin \theta (\frac{3}{4}{\cos }^2\theta -\frac{1}{4}{\sin }^2\theta )$

$=\frac{1}{4}(3\sin \theta -4{\sin }^3\theta )$

$=\frac{\sin 3\theta }{4}$

By putting $\theta ={20}^{\circ },$

$\sin {20}^{\circ }\sin {40}^{\circ }\sin {80}^{\circ }\sin {60}^{\circ }=\frac{\sin {60}^{\circ }}{4}\cdot \sin {60}^{\circ }$

$=\frac{3}{16}$