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Question

Mathematics Question on Trigonometric Functions

The value of sin⁡20∘  sin⁡40∘  sin⁡80∘\sin 20^{\circ} \; \sin 40^{\circ} \; \sin 80^{\circ} equals

A

32\frac{\sqrt{3}}{2}

B

34\frac{\sqrt{3}}{4}

C

38\frac{\sqrt{3}}{8}

D

18\frac{1}{8}

Answer

38\frac{\sqrt{3}}{8}

Explanation

Solution

We have, sin 20∘ sin 40∘ sin 80∘sin \,20^{\circ} \,sin \,40^{\circ}\, sin\, 80^{\circ}
=sin 20∘ sin(60∘−20∘)sin(60∘+20∘)= sin\, 20^{\circ}\, sin\left(60^{\circ} - 20^{\circ}\right) sin\left(60^{\circ}+20^{\circ}\right)
=14sin 3(20∘)=sin 60∘4=38= \frac{1}{4} sin \,3 \left(20^{\circ}\right) = \frac{sin\, 60^{\circ}}{4} = \frac{\sqrt{3}}{8}