Prove that Sin20° sin40° sin60° sin80° = 3/16
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Take sin40° and sin80° together
sin80°sin40°sin20°sin60°
2sin80°sin40°sin20°sin60°/2
(cos40°-cos120°)√3/2sin20°/2
√3/4(cos40°-cos120°)sin20°
√3/4(cos40°-(-1/2))sin20°
√3/4(cos40°+1/2)sin20°
√3/4((2cos40°+1)/2)sin20°
√3/8(2cos40°sin20°+sin20°)
[2cosasinb=sin(a+b)-sin(a-b)]
√3/8(sin60°-sin20°+sin20°)
√3/8*sin60°
√3/8*√3/2
3/16
sin80°sin40°sin20°sin60°
2sin80°sin40°sin20°sin60°/2
(cos40°-cos120°)√3/2sin20°/2
√3/4(cos40°-cos120°)sin20°
√3/4(cos40°-(-1/2))sin20°
√3/4(cos40°+1/2)sin20°
√3/4((2cos40°+1)/2)sin20°
√3/8(2cos40°sin20°+sin20°)
[2cosasinb=sin(a+b)-sin(a-b)]
√3/8(sin60°-sin20°+sin20°)
√3/8*sin60°
√3/8*√3/2
3/16
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