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

The general solution for cos(x)=3cos(x)-2 is x=2pin

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

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

x = 2 πn

Degrees

x = 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

cos (x) = u

u = 3 u - 2

u = 3 u - 2 : u = 1

u = 3 u - 2

Move

3 u

to the left side

u = 3 u - 2

Subtract

3 u

from both sides

u - 3 u = 3 u - 2 - 3 u

Simplify

- 2 u = - 2

- 2 u = - 2

Divide both sides by

- 2

- 2 u = - 2

Divide both sides by

- 2

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

Simplify

u = 1

u = 1

Substitute back

u = cos (x)

cos (x) = 1

cos (x) = 1

cos (x) = 1 : x = 2 πn

cos (x) = 1

General solutions for

cos (x) = 1

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 = 0 + 2 πn

x = 0 + 2 πn

Solve

x = 0 + 2 πn : x = 2 πn

x = 0 + 2 πn

0 + 2 πn = 2 πn

x = 2 πn

x = 2 πn

Combine all the solutions

x = 2 πn

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

2 sin (x) + 1 = csc (x)

tan^{2} (x) = - tan (x)

7 sin (x) + 3 = sin (x)

2 cos (2 x) = 3

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