What is the general solution for 6cos(2θ)=18cos(θ)-12 ?

The general solution for 6cos(2θ)=18cos(θ)-12 is θ=2pin,θ= pi/3+2pin,θ=(5pi)/3+2pin

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6cos(2θ)=18cos(θ)-12

6 cos (2 θ) = 18 cos (θ) - 12

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

Solution steps

6 cos (2 θ) = 18 cos (θ) - 12

Subtract

18 cos (θ) - 12

from both sides

6 cos (2 θ) - 18 cos (θ) + 12 = 0

Rewrite using trig identities

6 + 12 cos^{2} (θ) - 18 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

cos (\frac{2}{3} x - \frac{π}{2}) = 1

tan^{2} (x) sin (x) = sin (x), 0 \leq x \leq 2 π

- 3 sin (t) = 0

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

3 cos^{2} (θ) + 7 cos (θ) - 8 = 0

What is the general solution for 6cos(2θ)=18cos(θ)-12 ?
The general solution for 6cos(2θ)=18cos(θ)-12 is θ=2pin,θ= pi/3+2pin,θ=(5pi)/3+2pin