What is the general solution for (5sqrt(3))/3 cos(x)=1+sqrt(3)cos(x) ?

The general solution for (5sqrt(3))/3 cos(x)=1+sqrt(3)cos(x) is x= pi/6+2pin,x=(11pi)/6+2pin

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(5sqrt(3))/3 cos(x)=1+sqrt(3)cos(x)

Solution

\frac{5 3}{3} cos (x) = 1 + 3 cos (x)

Solution

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Degrees

x = 3 0^{\circ} + 36 0^{\circ} n, x = 33 0^{\circ} + 36 0^{\circ} n

Solution steps

\frac{5 3}{3} cos (x) = 1 + 3 cos (x)

Solve by substitution

\frac{5 3}{3} cos (x) = 1 + 3 cos (x)

Let:

cos (x) = u

\frac{5 3}{3} u = 1 + 3 u

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

\frac{5 3}{3} u = 1 + 3 u

Move

3 u

to the left side

\frac{5 3}{3} u = 1 + 3 u

Subtract

3 u

from both sides

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

Simplify

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

Simplify

\frac{5 3}{3} u - 3 u : \frac{2 3 u}{3}

\frac{5 3}{3} u - 3 u

Multiply

\frac{5 3}{3} u : \frac{5 3 u}{3}

\frac{5 3}{3} u

Multiply fractions:

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

= \frac{5 3 u}{3}

= \frac{5 3 u}{3} - 3 u

Convert element to fraction:

3 u = \frac{3 u 3}{3}

= \frac{5 3 u}{3} - \frac{3 u \cdot 3}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{5 3 u - 3 u \cdot 3}{3}

Add similar elements:

53 u - 33 u = 23 u

= \frac{2 3 u}{3}

Simplify

1 + 3 u - 3 u : 1

1 + 3 u - 3 u

Add similar elements:

3 u - 3 u = 0

= 1

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

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

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

Multiply both sides by

3

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

Multiply both sides by

3

\frac{3 \cdot 2 3 u}{3} = 1 \cdot 3

Simplify

23 u = 3

23 u = 3

Divide both sides by

23

23 u = 3

Divide both sides by

23

\frac{2 3 u}{2 3} = \frac{3}{2 3}

Simplify

\frac{2 3 u}{2 3} = \frac{3}{2 3}

Simplify

\frac{2 3 u}{2 3} : u

\frac{2 3 u}{2 3}

Divide the numbers:

\frac{2}{2} = 1

= \frac{3 u}{3}

Cancel the common factor:

3

= u

Simplify

\frac{3}{2 3} : \frac{3}{2}

\frac{3}{2 3}

Apply radical rule:

n a = a^{\frac{1}{n}}

3 = 3^{\frac{1}{2}}

= \frac{3}{2 \cdot 3 ^{\frac{1}{2}}}

Apply exponent rule:

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

\frac{3 ^{1}}{3 ^{\frac{1}{2}}} = 3^{1 - \frac{1}{2}}

= \frac{3 ^{1 - \frac{1}{2}}}{2}

Subtract the numbers:

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

= \frac{3 ^{\frac{1}{2}}}{2}

Apply radical rule:

a^{\frac{1}{n}} = n a

3^{\frac{1}{2}} = 3

= \frac{3}{2}

u = \frac{3}{2}

u = \frac{3}{2}

u = \frac{3}{2}

Substitute back

u = cos (x)

cos (x) = \frac{3}{2}

cos (x) = \frac{3}{2}

cos (x) = \frac{3}{2} : x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

cos (x) = \frac{3}{2}

General solutions for

cos (x) = \frac{3}{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}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Combine all the solutions

x = \frac{π}{6} + 2 πn, x = \frac{11 π}{6} + 2 πn

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for (5sqrt(3))/3 cos(x)=1+sqrt(3)cos(x) ?
The general solution for (5sqrt(3))/3 cos(x)=1+sqrt(3)cos(x) is x= pi/6+2pin,x=(11pi)/6+2pin