What is the general solution for 6cos(θ)=2+2cos(θ) ?

The general solution for 6cos(θ)=2+2cos(θ) is θ= pi/3+2pin,θ=(5pi)/3+2pin

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6cos(θ)=2+2cos(θ)

6 cos (θ) = 2 + 2 cos (θ)

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

Degrees

θ = 6 0^{\circ} + 36 0^{\circ} n, θ = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

6 cos (θ) = 2 + 2 cos (θ)

Solve by substitution

6 cos (θ) = 2 + 2 cos (θ)

Let:

cos (θ) = u

6 u = 2 + 2 u

6 u = 2 + 2 u : u = \frac{1}{2}

6 u = 2 + 2 u

Move

2 u

to the left side

6 u = 2 + 2 u

Subtract

2 u

from both sides

6 u - 2 u = 2 + 2 u - 2 u

Simplify

4 u = 2

4 u = 2

Divide both sides by

4

4 u = 2

Divide both sides by

4

\frac{4 u}{4} = \frac{2}{4}

Simplify

u = \frac{1}{2}

u = \frac{1}{2}

Substitute back

u = cos (θ)

cos (θ) = \frac{1}{2}

cos (θ) = \frac{1}{2}

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

cos (θ) = \frac{1}{2}

General solutions for

cos (θ) = \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}

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

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

Combine all the solutions

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

64 cos^{2} (x) = 49

x, sin (x) + sin (y) = 1

cos^{2} (x) - 1 = sin^{2} (x)

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

1 - sin (x) = 3 cos (x), 0 \leq x \leq 2 π

What is the general solution for 6cos(θ)=2+2cos(θ) ?
The general solution for 6cos(θ)=2+2cos(θ) is θ= pi/3+2pin,θ=(5pi)/3+2pin