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

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

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sqrt(3)cot(x/3)+1=0

Solution

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

Solution

x = 2 π + 3 πn

Degrees

x = 36 0^{\circ} + 54 0^{\circ} n

Solution steps

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

Move

1

to the right side

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

Subtract

1

from both sides

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

Simplify

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

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

Divide both sides by

3

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

Divide both sides by

3

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

Simplify

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

Simplify

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

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

Cancel the common factor:

3

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

Simplify

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

\frac{- 1}{3}

Apply the fraction rule:

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

= - \frac{1}{3}

Rationalize

- \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}

= - \frac{3}{3}

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

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

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

General solutions for

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

\frac{x}{3} = \frac{2 π}{3} + πn

Solve

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

\frac{x}{3} = \frac{2 π}{3} + πn

Multiply both sides by

3

\frac{x}{3} = \frac{2 π}{3} + πn

Multiply both sides by

3

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

Simplify

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

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

3 \cdot \frac{2 π}{3} + 3 πn : 2 π + 3 πn

3 \cdot \frac{2 π}{3} + 3 πn

3 \cdot \frac{2 π}{3} = 2 π

3 \cdot \frac{2 π}{3}

Multiply fractions:

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

= \frac{2 π 3}{3}

Cancel the common factor:

3

= 2 π

= 2 π + 3 πn

x = 2 π + 3 πn

x = 2 π + 3 πn

x = 2 π + 3 πn

x = 2 π + 3 πn

Popular Examples

sin(x)=0.1115

sin (x) = 0.1115

arccos(x)=11555

arccos (x) = 11555

sin(A)=-1/2

sin (A) = - \frac{1}{2}

6tan(x)+1=4tan(x)+3

6 tan (x) + 1 = 4 tan (x) + 3

cos(x)=(-sqrt(5))/5

cos (x) = \frac{- 5}{5}

Frequently Asked Questions (FAQ)

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