Question

value of sin pi/14 sin 3pi/14 sin 5pi/14

Answer 1

Given,
sin(π/14) .sin(3π/14). Sin(5π/14)
we should change some terms in cos for making easier .
sin(3π/14) = sin(π/2 - 4π/14) = cos(4π/14)
∴ sin(3π/14) = cos(4π/14)
Similarly, sin(5π/14) = sin(π/2 - 2π/14) = cos(2π/14)
Then, sin(π/14).sin(3π/14).sin(5π/14) = sin(π/14).cos(4π/14).cos(2π/14)

π/14 is heavy term to use again and again .
Let π/14 = x
then, sin(π/14).cos(4π/14).cos(2π/14) = sinx.cos4x.cos2x
Now, Let's start to solve ,

= Sinx.cos4x.cos2

multiply and divide by (2cosx)

= {2cosx.sinx.cos4x.cos2x}/(2cos2x)
= Sin2x.cos4x.cos2x/(2cos2x) [We know, sin2A = 2sinA.cosA ]
= {2sin2x.cos2x.cos4x}/(4cosx)
= sin4x.cos4x/(4cosx)
= (2sin4x.cos4x)/(8cosx)
= sin8x/8cosx
Put x = π/14
= sin8π/14/8cosπ/14
∵ sin8π/14 = sin(π/2 + π/14) = cosπ/14
so, sin8π/14/8cosπ/14 = cosπ/14/8cosπ/14 = 1/8

Hence, answer is 1/8