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

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

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Popular Trigonometry >

1/(cot^2(x))+(sqrt(3))/(cot(x))=0

Solution

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

Solution

x = \frac{2 π}{3} + πn

Degrees

x = 12 0^{\circ} + 18 0^{\circ} n

Solution steps

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

Solve by substitution

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

Let:

cot (x) = u

\frac{1}{u ^{2}} + \frac{3}{u} = 0

\frac{1}{u ^{2}} + \frac{3}{u} = 0 : u = - \frac{3}{3}

\frac{1}{u ^{2}} + \frac{3}{u} = 0

Multiply by LCM

\frac{1}{u ^{2}} + \frac{3}{u} = 0

Find Least Common Multiplier of

u^{2}, u : u^{2}

u^{2}, u

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear either in

u^{2}

u

= u^{2}

Multiply by LCM=

u^{2}

\frac{1}{u ^{2}} u^{2} + \frac{3}{u} u^{2} = 0 \cdot u^{2}

Simplify

\frac{1}{u ^{2}} u^{2} + \frac{3}{u} u^{2} = 0 \cdot u^{2}

Simplify

\frac{1}{u ^{2}} u^{2} : 1

\frac{1}{u ^{2}} u^{2}

Multiply fractions:

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

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

Cancel the common factor:

u^{2}

= 1

Simplify

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

\frac{3}{u} u^{2}

Multiply fractions:

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

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

Cancel the common factor:

u

= 3 u

Simplify

0 \cdot u^{2} : 0

0 \cdot u^{2}

Apply rule

0 \cdot a = 0

= 0

1 + 3 u = 0

1 + 3 u = 0

1 + 3 u = 0

Move

1

to the right side

1 + 3 u = 0

Subtract

1

from both sides

1 + 3 u - 1 = 0 - 1

Simplify

3 u = - 1

3 u = - 1

Divide both sides by

3

3 u = - 1

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 u}{3} : u

\frac{3 u}{3}

Cancel the common factor:

3

= u

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}

u = - \frac{3}{3}

u = - \frac{3}{3}

u = - \frac{3}{3}

Verify Solutions

Find undefined (singularity) points:

u = 0

Take the denominator(s) of

\frac{1}{u ^{2}} + \frac{3}{u}

and compare to zero

Solve

u^{2} = 0 : u = 0

u^{2} = 0

Apply rule

x^{n} = 0 \Rightarrow x = 0

u = 0

u = 0

The following points are undefined

u = 0

Combine undefined points with solutions:

u = - \frac{3}{3}

Substitute back

u = cot (x)

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

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

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

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

General solutions for

cot (x) = - \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

x = \frac{2 π}{3} + πn

x = \frac{2 π}{3} + πn

Combine all the solutions

x = \frac{2 π}{3} + πn

Graph

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Frequently Asked Questions (FAQ)

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