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Many-One onto Function

Prove sin 200...

Question

Prove $s i n 20^{0} s i n 40^{0} s i n 60^{0} s i n 60^{0} = \frac{3}{16}$

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Solution

$= s i n 20 s i n 40 \times \frac{\sqrt{3}}{2} \times s i n 80$
$= \frac{\sqrt{3}}{2} \times \frac{2}{2} s i n 20 s i n 40 s i n 80$
$= \frac{\sqrt{3}}{4} \times (2 s i n 20 s i n 40) s i n 80$
$= \frac{\sqrt{3}}{4} [c o s (20 - 40) - c o s (20 + 40)] s i n 80$
$= \frac{\sqrt{3}}{4} [c o s 20 - c o s 60] s i n 80$
$= \frac{\sqrt{3}}{4} [c o s 20 - \frac{1}{2}] s i n 80$
$= \frac{\sqrt{3}}{4} [c o s 20 s i n 80 - \frac{s i n 80}{2}]$
$= \frac{\sqrt{3}}{4} [\frac{2 s i n 80 c o s 20}{2} - \frac{s i n 80}{2}]$
$= \frac{\sqrt{3}}{4} [\frac{s i n (80 + 20) + s i n (80 - 20)}{2} - \frac{s i n 80}{2}]$
$= \frac{\sqrt{3}}{4} [\frac{s i n 100 + s i n 60}{2} - \frac{s i n 80}{2}]$
$= \frac{\sqrt{3}}{4} [\frac{s i n 100 - s i n 80}{2} + \frac{\sqrt{3}}{4}]$
$= \frac{\sqrt{3}}{4} [\frac{2 c o s (\frac{100 + 80}{2}) s i n (\frac{100 - 80}{2})}{2} + \frac{\sqrt{3}}{4}]$
$= \frac{\sqrt{3}}{4} [\frac{c o s 90 s i n 10}{2} + \frac{s q r t 3}{4}]$
$= \frac{\sqrt{3}}{4} [\frac{0 \times s i n 10}{2} + \frac{\sqrt{3}}{4}]$
$= \frac{\sqrt{3}}{4} \times \frac{\sqrt{3}}{4}$
$= \frac{3}{16}$

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