Question

evaluate:

$\sin 20\cdot \sin 40\cdot \sin 60.sin80$

Answer 1

$\sin 20°\sin 40°\sin 60°\sin 80°$

$=\sin (30-10)+\sin (30+10)\times \frac{\sqrt{3}}{2}\sin (90-10)$

$=\frac{\sqrt{3}}{2}\times (\sin 30\cos 10-\sin 10\cos 30)\times (\sin 30\cos 10+\sin 10\cos 30)\times (\sin 90\cos 10-\sin 10\cos 90)$

$=\frac{\sqrt{3}}{2}\times ({\sin }^{2}30{\cos }^{2}10-{\sin }^{2}10{\cos }^{2}30)\times \cos 10$

$=\frac{\sqrt{3}}{2}({\cos }^{2}10\times \frac{1}{4}-{\sin }^{2}10\times \frac{3}{4})\times \cos 10$

$=\frac{\sqrt{3}}{2}\times ({\cos }^{2}10\times \frac{1}{4}-(1-{\cos }^{2}10)\times \frac{3}{4})\times \cos 10$

$=\frac{\sqrt{3}}{2}\times (\frac{\left({\cos }^{2}10\right)}{4}-\frac{3}{4}+3\frac{\left({\cos }^{2}10\right)}{4})\times \cos 10$

$=\frac{\sqrt{3}}{2}\times ({\cos }^{2}10-\frac{3}{4})\times \cos 10$

$=\frac{\sqrt{3}}{2}\times \frac{(4{\cos }^{3}10-3\cos 10)}{4}$

$=\frac{\sqrt{3}}{2}\times \cos 30$

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

$=\frac{3}{16}$

$=0.1875$

Hence, the answer is $\mathrm{0.1875.}$