Question

Find the value of sin180/14 . sin3x180/14 . sin5x180/14 . sin7x180/14 ?

Comments

Apply Trigonometry Property, You will get the answer. You may ask if you have any problem

Answer 1

We know that:

sin(x) = cos(π/2−x) 
cos(−x) = cos(x) 
sin(2x) = 2 sinx cosx ---> sinx cosx = 1/2 sin(2x) 

LHS = sin(π/14) sin(3π/14) sin(5π/14) sin(7π/14) 
= sin(π/14) cos(4π/14) cos(2π/14) * 1
= sin(π/14) cos(2π/14) cos(4π/14) * cos(π/14)/cos(π/14) 
= cos(π/14) sin(π/14) cos(2π/14) cos(4π/14) / cos(π/14) 
= 1/2 sin(2π/14) cos(2π/14) cos(4π/14) / cos(π/14) 
= 1/2 * 1/2 sin(4π/14) cos(4π/14) / cos(π/14) 
= 1/2 * 1/2 * 1/2 sin(8π/14) / cos(π/14) 
= 1/2 * 1/2 * 1/2 cos(−π/14) / cos(π/14) 
= 1/2 * 1/2 * 1/2 cos(π/14) / cos(π/14) 
= 1/8