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

The general solution for 3cos(x)-sqrt(3)=cos(x) is x= pi/6+2pin,x=(11pi)/6+2pin

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3cos(x)-sqrt(3)=cos(x)

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

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Degrees

x = 3 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

cos (x) = u

3 u - 3 = u

3 u - 3 = u : u = \frac{3}{2}

3 u - 3 = u

Move

3

to the right side

3 u - 3 = u

Add

3

to both sides

3 u - 3 + 3 = u + 3

Simplify

3 u = u + 3

3 u = u + 3

Move

u

to the left side

3 u = u + 3

Subtract

u

from both sides

3 u - u = u + 3 - u

Simplify

2 u = 3

2 u = 3

Divide both sides by

2

2 u = 3

Divide both sides by

2

\frac{2 u}{2} = \frac{3}{2}

Simplify

u = \frac{3}{2}

u = \frac{3}{2}

Substitute back

u = cos (x)

cos (x) = \frac{3}{2}

cos (x) = \frac{3}{2}

cos (x) = \frac{3}{2} : x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

cos (x) = \frac{3}{2}

General solutions for

cos (x) = \frac{3}{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{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

sin (x) - 1 = - cos (x)

36 cos (2 x) = 0

2 sin^{2} (θ) + sin (θ) = 0, 0 \leq θ < 2 π

sin (x) + cos (x) = 0, - π \leq x \leq π

cos (θ) = \frac{- 1}{2}, 0 \leq θ \leq 4 π

What is the general solution for 3cos(x)-sqrt(3)=cos(x) ?
The general solution for 3cos(x)-sqrt(3)=cos(x) is x= pi/6+2pin,x=(11pi)/6+2pin