# The value of sin20.sin40.sin60.sin80 is?

Aug 10, 2018

$\sin {20}^{\circ} \sin {40}^{\circ} \sin {60}^{\circ} \sin {80}^{\circ} = \frac{3}{16}$

#### Explanation:

Let ,

$X = \sin {20}^{\circ} \sin {40}^{\circ} \sin {60}^{\circ} \sin {80}^{\circ}$

$\therefore X = \frac{1}{2} \left(\sin {60}^{\circ} \sin {20}^{\circ}\right) \left(2 \sin {80}^{\circ} \sin {40}^{\circ}\right)$

Using ,$\text{Product Identity}$

$2 \sin x \sin y = \cos \left(x - y\right) - \cos \left(x + y\right) , f \mathmr{and} , x = 80 , y = 40$

$\therefore X = \frac{1}{2} \left(\frac{\sqrt{3}}{2} \sin {20}^{\circ}\right) \left\{\cos \left(80 - 40\right) - \cos \left(80 + 40\right)\right\}$

$\therefore X = \frac{\sqrt{3}}{4} \sin {20}^{\circ} \left\{\cos {40}^{\circ} - \cos {120}^{\circ}\right\}$

$\therefore X = \frac{\sqrt{3}}{4} \sin {20}^{\circ} \left\{\cos {40}^{\circ} + \frac{1}{2}\right\}$

$\therefore X = \frac{\sqrt{3}}{8} \left\{2 \sin {20}^{\circ} \cos 40 + \sin 20\right\}$

Using ,$\text{Product Identity}$

$2 \sin x \cos y = \sin \left(x + y\right) + \sin \left(x - y\right) , f \mathmr{and} , x = 20 , y = 40$

$\therefore X = \frac{\sqrt{3}}{8} \left\{\sin 60 + \sin \left(- 20\right) + \sin 20\right\}$

$\therefore X = \frac{\sqrt{3}}{8} \left\{\frac{\sqrt{3}}{2} - \sin {20}^{\circ} + \sin {20}^{\circ}\right\}$

$\therefore X = \frac{\sqrt{3}}{8} \left\{\frac{\sqrt{3}}{2}\right\}$

$\therefore X = \frac{3}{16}$