What is the general solution for 2cos(2x)+3=4cos(x) ?

The general solution for 2cos(2x)+3=4cos(x) is x= pi/3+2pin,x=(5pi)/3+2pin

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2cos(2x)+3=4cos(x)

2 cos (2 x) + 3 = 4 cos (x)

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

Solution steps

2 cos (2 x) + 3 = 4 cos (x)

Subtract

4 cos (x)

from both sides

2 cos (2 x) + 3 - 4 cos (x) = 0

Rewrite using trig identities

1 - 4 cos (x) + 4 cos^{2} (x) = 0

Solve by substitution

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

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

Combine all the solutions

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

sin (θ) = \frac{1}{4}, cos (θ)

2 sin^{2} (β) = 9 sin (β) - 4

4 cos^{2} (x) - 1 = 4, cos (2 x) - 1 = 0

sin (2 θ) = 0.4

0.5 = sin (x) cos (x)

What is the general solution for 2cos(2x)+3=4cos(x) ?
The general solution for 2cos(2x)+3=4cos(x) is x= pi/3+2pin,x=(5pi)/3+2pin