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

The general solution for -6cos(θ)=2-2cos(θ) is θ=(2pi)/3+2pin,θ=(4pi)/3+2pin

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Popular >

-6cos(θ)=2-2cos(θ)

Solution

- 6 cos (θ) = 2 - 2 cos (θ)

Solution

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

Degrees

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

Add

2 u

to 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

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

Simplify

\frac{- 4 u}{- 4} : u

\frac{- 4 u}{- 4}

Apply the fraction rule:

\frac{- a}{- b} = \frac{a}{b}

= \frac{4 u}{4}

Divide the numbers:

\frac{4}{4} = 1

= u

Simplify

\frac{2}{- 4} : - \frac{1}{2}

\frac{2}{- 4}

Apply the fraction rule:

\frac{a}{- b} = - \frac{a}{b}

= - \frac{2}{4}

Cancel the common factor:

2

= - \frac{1}{2}

u = - \frac{1}{2}

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

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

Combine all the solutions

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

Popular Examples

cot^2(θ)-1=csc^2(θ)

cot^{2} (θ) - 1 = csc^{2} (θ)

sec(4x)-sec(2x)=2

sec (4 x) - sec (2 x) = 2

-cos(x)=1

- cos (x) = 1

cot(x)sin(x)-sin(x)=0

cot (x) sin (x) - sin (x) = 0

1.16cos(θ)+0.532cos(θ)-0.557=0

1.16 cos (θ) + 0.532 cos (θ) - 0.557 = 0

Frequently Asked Questions (FAQ)

What is the general solution for -6cos(θ)=2-2cos(θ) ?
The general solution for -6cos(θ)=2-2cos(θ) is θ=(2pi)/3+2pin,θ=(4pi)/3+2pin