Question

The general solution of equation sinx+sin5x=sin2x+sin4x is?

Answer 1

f(x) = sin x + 5x = sin 2x + sin 4x
Use trig identity:
`sin a + sin b = 2sin ((a + b)/2)cos ((a - b)/2`
Left side:
sin x + sin 5x = 2sin 3x.cos 2x
Right side:
sin 2x + sin 4x = 2sin 3x.cos x
f(x) = 2sin 3x.cos 2x - 2sin 3x.cos x = 0
(2sin 3x)(cos 2x - cos x) = 0
Either factor should be zero.
a. sin 3x = 0
`x = 2kpi`, and `x = pi + 2kpi = (2k+1)pi`, and `x = 2kpi`
b. cos 2x - cos x = 0
Reminder of trig identity:
`cos a - cos b = - 2sin ((a + b)/2).sin ((a - b)/2)`
In this case:
`cos 2x - cos x = - 2sin (3x/2).sin (x/2)`
a. `sin (3x/2) = 0` -->
`(3x/2) = 2kpi` --> `x = (4kpi)/3`
`(3x/2) = pi + 2kpi = (2k + 1)pi` --> `x = (2k + 1)(2pi/3)`
b. `sin (x/2) = 0`
`x/2 = 2kpi`--> `x = 4kpi = 2npi`
`x/2 = (2k +1)pi` --> `x = (2k +1)2pi = 2npi`