InterviewSolution
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InterviewSolution
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\operatorname sin \frac \pi 14 \operatorname sin \frac 3 \pi 14 \operatorname sin \frac 5 \pi 14 |
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Answer» Use sin x =cos(pi/2 -x) and cos x = sin(pi/2 + x),sin(pi/14) = cos(6pi/14), sin(3pi/14)=cos(4pi/14), sin(5pi/14)=cos(2pi/14), sin(7pi/14)=1,sin(9pi/14)=sin(pi/2 + 2pi/14)=cos(2pi/14), sin(11pi/14)=cos(4pi/14) and sin(13pi/14)=cos(6pi/14). So, total product = [cos(2pi/14) * cos(4pi/14) * cos(6pi/14)] ^ 2.Multiply and divide by sin2(2pi/14),product = [sin(2pi/14) * cos(2pi/14) * cos(4pi/14) * cos(6pi/14) / sin(2pi/14)] ^ 2,Use sin2x = 2*sin(x)*cos(x),product = ¼ * [sin(4pi/14) * cos(4pi/14) * cos(6pi/14) / sin(2pi/14) ] ^ 2,use sin(2x) identity again,product = 1/16 * [sin(8pi/14) * cos(6pi/14) / sin(2pi/14) ] ^2,Use [ 2 * sin a * cos b = sin(a+b) + sin(a-b) ],product = 1/64 * [ (sin(14pi/14) + sin(2pi/14))/sin(2pi/14) ] ^2,sin(14pi/14) =0 and sin(2pi/14) will get cancelled,So,Product = 1/64 |
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