What is the general solution for 2+cos(2x)=3cos(x) ?

The general solution for 2+cos(2x)=3cos(x) is x=2pin,x= pi/3+2pin,x=(5pi)/3+2pin

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

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

x = 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Solution steps

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

Subtract

3 cos (x)

from both sides

2 + cos (2 x) - 3 cos (x) = 0

Rewrite using trig identities

1 + 2 cos^{2} (x) - 3 cos (x) = 0

Solve by substitution

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

cos (x) = 1 : x = 2 πn

cos (x) = \frac{1}{2} : x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Combine all the solutions

x = 2 πn, x = \frac{π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

3 cot (θ) + 4 = 0

2 sin (3 θ) = 0

cot (t) = 3

3 sin (2 x) = 2

x, cos (2 x) = 0

What is the general solution for 2+cos(2x)=3cos(x) ?
The general solution for 2+cos(2x)=3cos(x) is x=2pin,x= pi/3+2pin,x=(5pi)/3+2pin