Question

evaluate Sin π/14 Sin 3π/14 Sin 5π/14 Sin9π/14 Sin 11π/14 sin 13π/14​

Answer 1

Answer:

(4) 1/64

Solution:

sin π/14 sin 3π/14 sin 5π/14 sin 7π/14 sin 9π/14 sin 11π/14 sin 13π/14

= sin π/14 sin 3π/14 sin 5π/14 (sin π/2) sin(π – 5π/14) sin(π – 3π/14) sin(π – π/14)

= sin π/14 sin 3π/14 sin 5π/14 (1) sin 5π/14 sin 3π/14 sin π/14

= sin2(π/14) sin2(3π/14) sin2(5π/14)

= [sin π/14 sin 3π/14 sin 5π/14]2

= [cos(π/2 – π/14) cos(π/2 – 3π/14) cos(π/2 – 5π/14)]2

= [cos 3π/7 cos 2π/7 cos π/7]2

= [-cos π/7 cos 2π/7 cos 4π/7]2

= [-(sin 23 π/7)/ (23 sin π/7)]2

= [-(1/8) {sin 8π/7}/ sin π/7]2

= [(1/8) sin π/7/ (sin π/7)]2

= (1/8)2

= 1/64

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