What is the general solution for cos(2θ)=3cos(θ)-2 ?

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

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cos(2θ)=3cos(θ)-2

cos (2 θ) = 3 cos (θ) - 2

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

Solution steps

cos (2 θ) = 3 cos (θ) - 2

Subtract

3 cos (θ) - 2

from both sides

cos (2 θ) - 3 cos (θ) + 2 = 0

Rewrite using trig identities

1 + 2 cos^{2} (θ) - 3 cos (θ) = 0

Solve by substitution

cos (θ) = 1, cos (θ) = \frac{1}{2}

cos (θ) = 1 : θ = 2 πn

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

Combine all the solutions

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

sin (x) = - \frac{5}{13}

9 tan^{2} (x) - 3 = 0

cos (x) = - \frac{2}{5}

cot^{2} (θ) - 9 = 0

sec (x) = - 1 - tan (x)

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