What is the general solution for cos(3x)=4cos(3x)-3cos(x) ?

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

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cos(3x)=4cos(3x)-3cos(x)

cos (3 x) = 4 cos (3 x) - 3 cos (x)

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

Solution steps

cos (3 x) = 4 cos (3 x) - 3 cos (x)

Subtract

4 cos (3 x) - 3 cos (x)

from both sides

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

Rewrite using trig identities

12 cos (x) - 12 cos^{3} (x) = 0

Solve by substitution

cos (x) = 0, cos (x) = - 1, cos (x) = 1

cos (x) = 0 : x = \frac{π}{2} + 2 πn, x = \frac{3 π}{2} + 2 πn

cos (x) = - 1 : x = π + 2 πn

cos (x) = 1 : x = 2 πn

Combine all the solutions

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

8 arctan (x) = 2 π

sin^{2} (θ) - 1 = cos^{2} (θ)

sin (\frac{1}{x}) = 0, 5

tan (2 x) sin (x) = sin (x)

sin (u) = - \frac{3}{5}, \frac{3 π}{2} < u < 2 π

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