Question

SOLVE THE EQUATION sinx+sin3x+sin5x

Answer 1

Use the following sum-to-product formula: 
sin(A) + sin(B) = 2sin[(A + B)/2]cos[(A - B)/2]. 

So: 
sin(5x) + sin(x) = 2sin[(5x + x)/2]cos[(5x - x)/2] 
= 2sin(3x)cos(2x). 

The equation now becomes: 
sin(x) + sin(3x) + sin(5x) = 0 
==> 2sin(3x)cos(2x) + sin(3x) = 0 
==> sin(3x)[2cos(2x) + 1] = 0, by factoring out sin(3x) 
==> sin(3x) = 0 and cos(2x) = -1/2, by the zero-product property. 

Using the unit circle: 
(a) sin(3x) = 0: 
3x = ±πk ==> x = ±πk/3 
(b) cos(2x) = -1/2: 
2x = 4π/3 ± 2πk and 2x = 7π/3 ± 2πk ==> x = 2π/3 ± πk and x = 7π/6 ± πk, 

where k is an integer. 

I hope this helps!