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  4. The value of sin 20° sin 40° sin 60° sin 80° is

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Q. The value of $\sin \, 20^{\circ} \, \sin \, 40^{\circ} \sin \, 60^{\circ} \, \sin \, 80^{\circ}$ is

Trigonometric Functions

Solution:

Indeed $\sin \, 20^{\circ} \, \sin \, 40^{\circ} \, \sin \, 60^{\circ} \, \sin \, 80^{\circ}$.
$ = \frac{\sqrt{3}}{2} \sin \, 20^{\circ} \sin (60^{\circ} - 20^{\circ} ) \sin (60^{\circ} + 20^{\circ} )$
(since $\sin 60^{\circ} = \frac{\sqrt{3}}{2} $)
$ = \frac{\sqrt{3}}{2} \sin20^{\circ} \left[\sin^{2} 60^{\circ} - \sin^{2}20^{\circ}\right]$
$= \frac{\sqrt{3}}{2} \sin 20^{\circ} \left[\frac{3}{4} - \sin^{2} 20^{\circ}\right] $
$ = \frac{\sqrt{3}}{2} \times\frac{1}{4} \left[3 \sin20^{\circ} - 4 \sin^{3} 20^{\circ}\right] $
$= \frac{\sqrt{3}}{2} \times\frac{1}{4} \left(\sin60^{\circ}\right) $
$= \frac{\sqrt{3}}{2} \times \frac{1}{4} \times\frac{\sqrt{3}}{2} = \frac{3}{16}$