What is the general solution for 4cos(x)+1=2cos(x) ?

The general solution for 4cos(x)+1=2cos(x) is x=(2pi)/3+2pin,x=(4pi)/3+2pin

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4cos(x)+1=2cos(x)

4 cos (x) + 1 = 2 cos (x)

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

Solution steps

4 cos (x) + 1 = 2 cos (x)

Solve by substitution

cos (x) = - \frac{1}{2}

cos (x) = - \frac{1}{2} : x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

Combine all the solutions

x = \frac{2 π}{3} + 2 πn, x = \frac{4 π}{3} + 2 πn

cos (2 x) + cos (4 x) + cos (6 x) = 0

2 cos (x) sin (x) - cos (x) = 0

6 sin (x) = 6 sin (2 x)

sin (5 x) + sin (x) = 3 cos (2 x)

0 = - 5 sin (2 x)

What is the general solution for 4cos(x)+1=2cos(x) ?
The general solution for 4cos(x)+1=2cos(x) is x=(2pi)/3+2pin,x=(4pi)/3+2pin