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

1 Answer
Nghi N.
Jun 22, 2018

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#

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