What is the general solution for 5cos(x)-3=9cos(x)-5 ?

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

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5cos(x)-3=9cos(x)-5

Solution

5 cos (x) - 3 = 9 cos (x) - 5

Solution

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

Degrees

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

Solution steps

5 cos (x) - 3 = 9 cos (x) - 5

Solve by substitution

5 cos (x) - 3 = 9 cos (x) - 5

Let:

cos (x) = u

5 u - 3 = 9 u - 5

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

5 u - 3 = 9 u - 5

Move

3

to the right side

5 u - 3 = 9 u - 5

Add

3

to both sides

5 u - 3 + 3 = 9 u - 5 + 3

Simplify

5 u = 9 u - 2

5 u = 9 u - 2

Move

9 u

to the left side

5 u = 9 u - 2

Subtract

9 u

from both sides

5 u - 9 u = 9 u - 2 - 9 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 (x)

cos (x) = \frac{1}{2}

cos (x) = \frac{1}{2}

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

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

Combine all the solutions

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

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Frequently Asked Questions (FAQ)

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