What is the general solution for sqrt(3)cot((3pi)/7 x)+3=0 ?

The general solution for sqrt(3)cot((3pi)/7 x)+3=0 is x= 35/18+(7n)/3

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sqrt(3)cot((3pi)/7 x)+3=0

Solution

3 cot (\frac{3 π}{7} x) + 3 = 0

Solution

x = \frac{35}{18} + \frac{7 n}{3}

Degrees

x = 111.40846 \dots^{\circ} + 133.69015 \dots^{\circ} n

Solution steps

3 cot (\frac{3 π}{7} x) + 3 = 0

Move

3

to the right side

3 cot (\frac{3 π}{7} x) + 3 = 0

Subtract

3

from both sides

3 cot (\frac{3 π}{7} x) + 3 - 3 = 0 - 3

Simplify

3 cot (\frac{3 π}{7} x) = - 3

3 cot (\frac{3 π}{7} x) = - 3

Divide both sides by

3

3 cot (\frac{3 π}{7} x) = - 3

Divide both sides by

3

\frac{3 cot ( \frac{3 π}{7} x )}{3} = \frac{- 3}{3}

Simplify

\frac{3 cot ( \frac{3 π}{7} x )}{3} = \frac{- 3}{3}

Simplify

\frac{3 cot ( \frac{3 π}{7} x )}{3} : cot (\frac{3 π}{7} x)

\frac{3 cot ( \frac{3 π}{7} x )}{3}

Cancel the common factor:

3

= cot (\frac{3 π}{7} x)

Simplify

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

\frac{- 3}{3}

Apply the fraction rule:

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

= - \frac{3}{3}

Apply radical rule:

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

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

= \frac{3}{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}}

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

Subtract the numbers:

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

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

Apply radical rule:

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

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

= - 3

cot (\frac{3 π}{7} x) = - 3

cot (\frac{3 π}{7} x) = - 3

cot (\frac{3 π}{7} x) = - 3

General solutions for

cot (\frac{3 π}{7} x) = - 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{3 π}{7} x = \frac{5 π}{6} + πn

\frac{3 π}{7} x = \frac{5 π}{6} + πn

Solve

\frac{3 π}{7} x = \frac{5 π}{6} + πn : x = \frac{35}{18} + \frac{7 n}{3}

\frac{3 π}{7} x = \frac{5 π}{6} + πn

Multiply both sides by

7

\frac{3 π}{7} x = \frac{5 π}{6} + πn

Multiply both sides by

7

7 \cdot \frac{3 π}{7} x = 7 \cdot \frac{5 π}{6} + 7 πn

Simplify

7 \cdot \frac{3 π}{7} x = 7 \cdot \frac{5 π}{6} + 7 πn

Simplify

7 \cdot \frac{3 π}{7} x : 3 π x

7 \cdot \frac{3 π}{7} x

Multiply fractions:

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

= \frac{3 \cdot 7 π}{7} x

Cancel the common factor:

7

= x \cdot 3 π

Simplify

7 \cdot \frac{5 π}{6} + 7 πn : \frac{35 π}{6} + 7 πn

7 \cdot \frac{5 π}{6} + 7 πn

7 \cdot \frac{5 π}{6} = \frac{35 π}{6}

7 \cdot \frac{5 π}{6}

Multiply fractions:

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

= \frac{5 π 7}{6}

Multiply the numbers:

5 \cdot 7 = 35

= \frac{35 π}{6}

= \frac{35 π}{6} + 7 πn

3 π x = \frac{35 π}{6} + 7 πn

3 π x = \frac{35 π}{6} + 7 πn

3 π x = \frac{35 π}{6} + 7 πn

Divide both sides by

3 π

3 π x = \frac{35 π}{6} + 7 πn

Divide both sides by

3 π

\frac{3 π x}{3 π} = \frac{\frac{35 π}{6}}{3 π} + \frac{7 πn}{3 π}

Simplify

\frac{3 π x}{3 π} = \frac{\frac{35 π}{6}}{3 π} + \frac{7 πn}{3 π}

Simplify

\frac{3 π x}{3 π} : x

\frac{3 π x}{3 π}

Divide the numbers:

\frac{3}{3} = 1

= \frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{\frac{35 π}{6}}{3 π} + \frac{7 πn}{3 π} : \frac{35}{18} + \frac{7 n}{3}

\frac{\frac{35 π}{6}}{3 π} + \frac{7 πn}{3 π}

\frac{\frac{35 π}{6}}{3 π} = \frac{35}{18}

\frac{\frac{35 π}{6}}{3 π}

Apply the fraction rule:

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

= \frac{35 π}{6 \cdot 3 π}

Multiply the numbers:

6 \cdot 3 = 18

= \frac{35 π}{18 π}

Cancel the common factor:

π

= \frac{35}{18}

\frac{7 πn}{3 π} = \frac{7 n}{3}

\frac{7 πn}{3 π}

Cancel the common factor:

π

= \frac{7 n}{3}

= \frac{35}{18} + \frac{7 n}{3}

x = \frac{35}{18} + \frac{7 n}{3}

x = \frac{35}{18} + \frac{7 n}{3}

x = \frac{35}{18} + \frac{7 n}{3}

x = \frac{35}{18} + \frac{7 n}{3}

Graph

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Frequently Asked Questions (FAQ)

What is the general solution for sqrt(3)cot((3pi)/7 x)+3=0 ?
The general solution for sqrt(3)cot((3pi)/7 x)+3=0 is x= 35/18+(7n)/3