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

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

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

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

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

Solution steps

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

Subtract

- 3 cos (x) - 2

from both sides

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

Rewrite using trig identities

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

Solve by substitution

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

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

cos (x) = - 1 : x = π + 2 πn

Combine all the solutions

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

12 sin^{2} (t) + cos (t) - 1 = 5

sin (x) = \frac{5}{12}

arctan (x) = 2

tan (2 x) = \frac{1}{3}

2 sin (θ) cos (θ) - 3 cos (θ) = 0

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