What is the general solution for 2-3cos(x)=5+3cos(x) ?

The general solution for 2-3cos(x)=5+3cos(x) is x=(2pi)/3+2pin,x=(4pi)/3+2pin

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2-3cos(x)=5+3cos(x)

Solution

2 - 3 cos (x) = 5 + 3 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 - 3 cos (x) = 5 + 3 cos (x)

Solve by substitution

2 - 3 cos (x) = 5 + 3 cos (x)

Let:

cos (x) = u

2 - 3 u = 5 + 3 u

2 - 3 u = 5 + 3 u : u = - \frac{1}{2}

2 - 3 u = 5 + 3 u

Move

2

to the right side

2 - 3 u = 5 + 3 u

Subtract

2

from both sides

2 - 3 u - 2 = 5 + 3 u - 2

Simplify

- 3 u = 3 u + 3

- 3 u = 3 u + 3

Move

3 u

to the left side

- 3 u = 3 u + 3

Subtract

3 u

from both sides

- 3 u - 3 u = 3 u + 3 - 3 u

Simplify

- 6 u = 3

- 6 u = 3

Divide both sides by

- 6

- 6 u = 3

Divide both sides by

- 6

\frac{- 6 u}{- 6} = \frac{3}{- 6}

Simplify

\frac{- 6 u}{- 6} = \frac{3}{- 6}

Simplify

\frac{- 6 u}{- 6} : u

\frac{- 6 u}{- 6}

Apply the fraction rule:

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

= \frac{6 u}{6}

Divide the numbers:

\frac{6}{6} = 1

= u

Simplify

\frac{3}{- 6} : - \frac{1}{2}

\frac{3}{- 6}

Apply the fraction rule:

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

= - \frac{3}{6}

Cancel the common factor:

3

= - \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

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Frequently Asked Questions (FAQ)

What is the general solution for 2-3cos(x)=5+3cos(x) ?
The general solution for 2-3cos(x)=5+3cos(x) is x=(2pi)/3+2pin,x=(4pi)/3+2pin