What is the general solution for 3cos(x)+9/2 =5+4cos(x) ?

The general solution for 3cos(x)+9/2 =5+4cos(x) is x=(2pi)/3+2pin,x=(4pi)/3+2pin

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3cos(x)+9/2 =5+4cos(x)

Solution

3 cos (x) + \frac{9}{2} = 5 + 4 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

3 cos (x) + \frac{9}{2} = 5 + 4 cos (x)

Solve by substitution

3 cos (x) + \frac{9}{2} = 5 + 4 cos (x)

Let:

cos (x) = u

3 u + \frac{9}{2} = 5 + 4 u

3 u + \frac{9}{2} = 5 + 4 u : u = - \frac{1}{2}

3 u + \frac{9}{2} = 5 + 4 u

Move

\frac{9}{2}

to the right side

3 u + \frac{9}{2} = 5 + 4 u

Subtract

\frac{9}{2}

from both sides

3 u + \frac{9}{2} - \frac{9}{2} = 5 + 4 u - \frac{9}{2}

Simplify

3 u + \frac{9}{2} - \frac{9}{2} = 5 + 4 u - \frac{9}{2}

Simplify

3 u + \frac{9}{2} - \frac{9}{2} : 3 u

3 u + \frac{9}{2} - \frac{9}{2}

Add similar elements:

\frac{9}{2} - \frac{9}{2} = 0

= 3 u

Simplify

5 + 4 u - \frac{9}{2} : 4 u + \frac{1}{2}

5 + 4 u - \frac{9}{2}

Combine the fractions

5 - \frac{9}{2} : \frac{1}{2}

5 - \frac{9}{2}

Convert element to fraction:

5 = \frac{5 \cdot 2}{2}

= \frac{5 \cdot 2}{2} - \frac{9}{2}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{5 \cdot 2 - 9}{2}

5 \cdot 2 - 9 = 1

5 \cdot 2 - 9

Multiply the numbers:

5 \cdot 2 = 10

= 10 - 9

Subtract the numbers:

10 - 9 = 1

= 1

= \frac{1}{2}

= 4 u + \frac{1}{2}

3 u = 4 u + \frac{1}{2}

3 u = 4 u + \frac{1}{2}

3 u = 4 u + \frac{1}{2}

Move

4 u

to the left side

3 u = 4 u + \frac{1}{2}

Subtract

4 u

from both sides

3 u - 4 u = 4 u + \frac{1}{2} - 4 u

Simplify

- u = \frac{1}{2}

- u = \frac{1}{2}

Divide both sides by

- 1

- u = \frac{1}{2}

Divide both sides by

- 1

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

Simplify

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

Simplify

\frac{- u}{- 1} : u

\frac{- u}{- 1}

Apply the fraction rule:

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

= \frac{u}{1}

Apply rule

\frac{a}{1} = a

= u

Simplify

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

\frac{\frac{1}{2}}{- 1}

Apply the fraction rule:

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

= - \frac{\frac{1}{2}}{1}

Apply the fraction rule:

\frac{a}{1} = a

\frac{\frac{1}{2}}{1} = \frac{1}{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

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Frequently Asked Questions (FAQ)

What is the general solution for 3cos(x)+9/2 =5+4cos(x) ?
The general solution for 3cos(x)+9/2 =5+4cos(x) is x=(2pi)/3+2pin,x=(4pi)/3+2pin