Maharashtra State BoardHSC Science (General) 11th
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Prove the following : sin 20° sin 40° sin 60° sin 80° = 316 - Mathematics and Statistics

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Sum

Prove the following :

sin 20° sin 40° sin 60° sin 80° = `3/16`

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Solution

L.H.S. = sin 20°· sin 40°· sin 60°· sin 80°

= `sqrt(3)/2*sin20^circ*  sin40^circ *  sin80^circ  .....[because sin60^circ = sqrt(3)/2]`

= `sqrt(3)/4(2  sin40^circ*  sin20^circ)*sin80^circ`

= `sqrt(3)/4[cos(40^circ - 20^circ) - cos(40^circ + 20^circ)] xx sin80^circ`

= `sqrt(3)/4[cos20^circ -  cos60^circ]*sin80^circ`

= `sqrt(3)/8[2  sin80^circ*  cos20^circ -  2  cos 60^circ*  sin80^circ]`

= `sqrt(3)/8[sin(80^circ + 20^circ) + sin(80^circ - 20^circ) - 2 xx 1/2*  sin80^circ]`

= `sqrt(3)/8[sin100^circ +  sin60^circ -  sin80^circ]`

= `sqrt(3)/8[sin(180^circ - 80^circ) + sqrt(3)/2 - sin80^circ]`

= `sqrt(3)/8(sin80^circ + sqrt(3)/2 - sin80^circ)`

= `sqrt(3)/8 xx sqrt(3)/2`

= `3/16`

= R.H.S.

Concept: Factorization Formulae - Formulae for Conversion of Product in to Sum Or Difference
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