Question

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

Answer 1

#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`