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

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

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

Solution

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

Simplify

\frac{1}{cot ^{2} ( x )} + 3 tan (x) : \frac{1 + 3 cot ^{2} ( x ) tan ( x )}{cot ^{2} ( x )}

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

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

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

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

Rewrite using trig identities

1 + cot^{2} (x) 3 tan (x)

Use the basic trigonometric identity:

tan (x) = \frac{1}{cot ( x )}

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

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

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

Multiply fractions:

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

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

Multiply:

1 \cdot cot^{2} (x) = cot^{2} (x)

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

Cancel the common factor:

cot (x)

= 3 cot (x)

= 1 + 3 cot (x)

1 + cot (x) 3 = 0

Move

1

to the right side

1 + cot (x) 3 = 0

Subtract

1

from both sides

1 + cot (x) 3 - 1 = 0 - 1

Simplify

cot (x) 3 = - 1

cot (x) 3 = - 1

Divide both sides by

3

cot (x) 3 = - 1

Divide both sides by

3

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

Simplify

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

Simplify

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

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

Cancel the common factor:

3

= cot (x)

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

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

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

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Frequently Asked Questions (FAQ)

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