What is the general solution for 4cos(x)+1=2cos(x) ?

The general solution for 4cos(x)+1=2cos(x) is x=(2pi)/3+2pin,x=(4pi)/3+2pin

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4cos(x)+1=2cos(x)

4 cos (x) + 1 = 2 cos (x)

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

Degrees

x = 12 0^{\circ} + 36 0^{\circ} n, x = 24 0^{\circ} + 36 0^{\circ} n

Solution steps

4 cos (x) + 1 = 2 cos (x)

Solve by substitution

4 cos (x) + 1 = 2 cos (x)

Let:

cos (x) = u

4 u + 1 = 2 u

4 u + 1 = 2 u : u = - \frac{1}{2}

4 u + 1 = 2 u

Move

1

to the right side

4 u + 1 = 2 u

Subtract

1

from both sides

4 u + 1 - 1 = 2 u - 1

Simplify

4 u = 2 u - 1

4 u = 2 u - 1

Move

2 u

to the left side

4 u = 2 u - 1

Subtract

2 u

from both sides

4 u - 2 u = 2 u - 1 - 2 u

Simplify

2 u = - 1

2 u = - 1

Divide both sides by

2

2 u = - 1

Divide both sides by

2

\frac{2 u}{2} = \frac{- 1}{2}

Simplify

u = - \frac{1}{2}

u = - \frac{1}{2}

Substitute back

u = cos (x)

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

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

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

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

General solutions for

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

cos (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

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

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

Combine all the solutions

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

cos (2 x) + cos (4 x) + cos (6 x) = 0

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

6 sin (x) = 6 sin (2 x)

sin (5 x) + sin (x) = 3 cos (2 x)

0 = - 5 sin (2 x)

What is the general solution for 4cos(x)+1=2cos(x) ?
The general solution for 4cos(x)+1=2cos(x) is x=(2pi)/3+2pin,x=(4pi)/3+2pin