Solve using the double angle understanding? Cos6x + cos4x +cos2x =0

1 Answer
Nghi N.
Jun 6, 2018

#pi/6; pi/4; pi/3; (3pi)/4; (+ (kpi)/2)

Explanation:

cos 6x + cos 4x + cos 2x = 0 (1)
Use trig identity:
#cos (a + b) = cos ((a + b)/2)cos ((a - b)/2)#
We have:
cos 6x + cos 2x = 2cos 4xcos 2x
Equation (1) becomes:
2cos 4x.cos 2x + cos 2x = 0
cos 2x(2cos 4x + 1) = 0
Either factor should be zero.
a. #cos 2x = 0#
Unit circle gives 2 solutions for 2x
1. #2x = pi/2 + 2kpi#, --> #x = pi/4 + kpi#
2. #2x = (3pi)/2 + 2kpi# --> #x = (3pi)/4 + kpi#
b. #2cos 4x + 1 = 0#
#cos 4x = - 1/2#
Trig table and unit circle give 2 solutions for 4x:
#4x = +- 2pi/3#
1. #4x = (2pi)/3 + 2kpi#--> #x = (pi/6) = (kpi)/2#
2. #4x = - (2pi)/3 = (4pi)/3# (co-terminal)-->
#x = (pi/3) + (kpi)/2#

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