s i n 20 ∘ s i n 40 ∘ s i n 60 ∘ s i n 80 ∘ = Options: A . &nbsp - 3 16 B . &nbsp 5 16 C . &nbsp 3 16 D . &nbsp - 5 16

Answer: Option C : C s i n 20 o s i n 40 ∘ s i n 60 ∘ s i n 80 ∘ = 1 2 s i n 20 ∘ s i n 60 ∘ ( 2 s i n 40 ∘ s i n 80 ∘ ) = 1 2 s i n 20 ∘ s i n 60 ∘ ( c o s 40 ∘ − c o s 120 ∘ ) = 1 2 . √ 3 2 s i n 20 ∘ ( 1 − 2 s i n 2 20 ∘ + 1 2 ) = √ 3 4 s i n 20 ∘ ( 3 2 − 2 s i n 2 20 ∘ ) = √ 3 8 ( 3 s i n 20 ∘ − 4 s i n 3 20 ∘ ) = √ 3 8 s i n 60 ∘ = √ 3 8 . √ 3 2 = 3 16

sin 20∘sin 40∘sin 60∘sin 80∘=

Discussion Forum : Trigonometric Functions

Question -

$s i n 20^{\circ} s i n 40^{\circ} s i n 60^{\circ} s i n 80^{\circ}$ =

Options:

A . -

\frac{3}{16}

B .

\frac{5}{16}

C .

\frac{3}{16}

D . -

\frac{5}{16}

Answer: Option C
:
C

$s i n 20^{o} s i n 40^{\circ} s i n 60^{\circ} s i n 80^{\circ}$
= $\frac{1}{2} s i n 20^{\circ} s i n 60^{\circ} (2 s i n 40^{\circ} s i n 80^{\circ})$
= $\frac{1}{2} s i n 20^{\circ} s i n 60^{\circ} (c o s 40^{\circ} - c o s 120^{\circ})$
= $\frac{1}{2} . \frac{\sqrt{3}}{2} s i n 20^{\circ} (1 - 2 s i n^{2} 20^{\circ} + \frac{1}{2})$
= $\frac{\sqrt{3}}{4} s i n 20^{\circ} (\frac{3}{2} - 2 s i n^{2} 20^{\circ})$
= $\frac{\sqrt{3}}{8} (3 s i n 20^{\circ} - 4 s i n^{3} 20^{\circ})$
= $\frac{\sqrt{3}}{8} s i n 60^{\circ} = \frac{\sqrt{3}}{8} . \frac{\sqrt{3}}{2} = \frac{3}{16}$

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Discussion Forum : Trigonometric Functions

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