Question

what is the value of this

Answer 1

sin(π/14) sin(3π/14) sin(5π/14)

Put : x = π/14.

Then : 14x = π.

∴ = sin x. sin 3x, sin 5x.

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

Now,

sin π/14 = sin (π/2 - 6π/14) = cos 6π/14 = cos (π - 8π/14) = - cos 8π/14 = - cos 8x

Similarly, sin 3π/14 = cos 4π/14 = cos 4x and

sin 5π/14 = cos 2π/14 = cos 2x.

∴ from (1),

= - cos 8x. cos 4x. cos 2x

Multiply and divide by 2 sin2x,

...=  -1/ (2 sin2x ). [( 2 sin 2x. cos 2x ). cos 4x. cos 8x]

...= (-1/ 4 sin 2x ). ( 2 sin 4x. cos 4x ). cos 8x

...= (-1/ 8 sin 2x ). ( 2 sin 8x. cos 8x )

...= (-1/ 8 sin 2x ). sin 16x

...= (-1/8). ( 1/ (sin π/7)). sin 8π/7

...= (-1/8). ( 1/ (sin π/7)). sin ( π + π/7 )

...= - (1/8). ( 1/ sin π/7 ). ( - sin π/7 )

...= 1/8.

∴ = 1/8 is the answer!!