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

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

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2-2cos(x)=-6cos(x)

Solution

2 - 2 cos (x) = - 6 cos (x)

Solution

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

2 - 2 cos (x) = - 6 cos (x)

Solve by substitution

2 - 2 cos (x) = - 6 cos (x)

Let:

cos (x) = u

2 - 2 u = - 6 u

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

2 - 2 u = - 6 u

Move

2

to the right side

2 - 2 u = - 6 u

Subtract

2

from both sides

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

Simplify

- 2 u = - 6 u - 2

- 2 u = - 6 u - 2

Move

6 u

to the left side

- 2 u = - 6 u - 2

Add

6 u

to both sides

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

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 (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

Popular Examples

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2 sin^{2} (u) = 1 + sin (u), 0 \leq t < 2 π

tan(3y+60)tan(2y+5)=1

tan (3 y + 60) tan (2 y + 5) = 1

18^2=8^2+20^2-2*8*20*cos(x)

1 8^{2} = 8^{2} + 2 0^{2} - 2 \cdot 8 \cdot 20 \cdot cos (x)

cot(θ)=-pi/6

cot (θ) = - \frac{π}{6}

3tan^2(x)-3tan(x)=0

3 tan^{2} (x) - 3 tan (x) = 0

Frequently Asked Questions (FAQ)

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