What is the general solution for sqrt(3)cot((2x)/3)-1=0 ?

The general solution for sqrt(3)cot((2x)/3)-1=0 is x= pi/2+(3pin)/2

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sqrt(3)cot((2x)/3)-1=0

Solution

3 cot (\frac{2 x}{3}) - 1 = 0

Solution

x = \frac{π}{2} + \frac{3 πn}{2}

Degrees

x = 9 0^{\circ} + 27 0^{\circ} n

Solution steps

3 cot (\frac{2 x}{3}) - 1 = 0

Move

1

to the right side

3 cot (\frac{2 x}{3}) - 1 = 0

Add

1

to both sides

3 cot (\frac{2 x}{3}) - 1 + 1 = 0 + 1

Simplify

3 cot (\frac{2 x}{3}) = 1

3 cot (\frac{2 x}{3}) = 1

Divide both sides by

3

3 cot (\frac{2 x}{3}) = 1

Divide both sides by

3

\frac{3 cot ( \frac{2 x}{3} )}{3} = \frac{1}{3}

Simplify

\frac{3 cot ( \frac{2 x}{3} )}{3} = \frac{1}{3}

Simplify

\frac{3 cot ( \frac{2 x}{3} )}{3} : cot (\frac{2 x}{3})

\frac{3 cot ( \frac{2 x}{3} )}{3}

Cancel the common factor:

3

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

Simplify

\frac{1}{3} : \frac{3}{3}

\frac{1}{3}

Multiply by the conjugate

\frac{3}{3}

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

1 \cdot 3 = 3

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= \frac{3}{3}

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

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

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

General solutions for

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

cot (x)

periodicity table with

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} cot (x) \mp \infty 31 \frac{3}{3} 0 - \frac{3}{3} - 1 - 3

\frac{2 x}{3} = \frac{π}{3} + πn

\frac{2 x}{3} = \frac{π}{3} + πn

Solve

\frac{2 x}{3} = \frac{π}{3} + πn : x = \frac{π}{2} + \frac{3 πn}{2}

\frac{2 x}{3} = \frac{π}{3} + πn

Multiply both sides by

3

\frac{2 x}{3} = \frac{π}{3} + πn

Multiply both sides by

3

\frac{3 \cdot 2 x}{3} = 3 \cdot \frac{π}{3} + 3 πn

Simplify

\frac{3 \cdot 2 x}{3} = 3 \cdot \frac{π}{3} + 3 πn

Simplify

\frac{3 \cdot 2 x}{3} : 2 x

\frac{3 \cdot 2 x}{3}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{6 x}{3}

Divide the numbers:

\frac{6}{3} = 2

= 2 x

Simplify

3 \cdot \frac{π}{3} + 3 πn : π + 3 πn

3 \cdot \frac{π}{3} + 3 πn

3 \cdot \frac{π}{3} = π

3 \cdot \frac{π}{3}

Multiply fractions:

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

= \frac{π 3}{3}

Cancel the common factor:

3

= π

= π + 3 πn

2 x = π + 3 πn

2 x = π + 3 πn

2 x = π + 3 πn

Divide both sides by

2

2 x = π + 3 πn

Divide both sides by

2

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

Simplify

x = \frac{π}{2} + \frac{3 πn}{2}

x = \frac{π}{2} + \frac{3 πn}{2}

x = \frac{π}{2} + \frac{3 πn}{2}

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Frequently Asked Questions (FAQ)

What is the general solution for sqrt(3)cot((2x)/3)-1=0 ?
The general solution for sqrt(3)cot((2x)/3)-1=0 is x= pi/2+(3pin)/2