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

The general solution for (cot(x)+1)(sqrt(3)tan(x)+1)=0 is x=(3pi)/4+pin,x=(5pi)/6+pin

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

Solution

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

Solution

x = \frac{3 π}{4} + πn, x = \frac{5 π}{6} + πn

Degrees

x = 13 5^{\circ} + 18 0^{\circ} n, x = 15 0^{\circ} + 18 0^{\circ} n

Solution steps

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

Solving each part separately

cot (x) + 1 = 0 or 3 tan (x) + 1 = 0

cot (x) + 1 = 0 : x = \frac{3 π}{4} + πn

cot (x) + 1 = 0

Move

1

to the right side

cot (x) + 1 = 0

Subtract

1

from both sides

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

Simplify

cot (x) = - 1

cot (x) = - 1

General solutions for

cot (x) = - 1

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{3 π}{4} + πn

x = \frac{3 π}{4} + πn

3 tan (x) + 1 = 0 : x = \frac{5 π}{6} + πn

3 tan (x) + 1 = 0

Move

1

to the right side

3 tan (x) + 1 = 0

Subtract

1

from both sides

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

Simplify

3 tan (x) = - 1

3 tan (x) = - 1

Divide both sides by

3

3 tan (x) = - 1

Divide both sides by

3

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

Simplify

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

Simplify

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

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

Cancel the common factor:

3

= tan (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}

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

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

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

General solutions for

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

tan (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} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{5 π}{6} + πn

x = \frac{5 π}{6} + πn

Combine all the solutions

x = \frac{3 π}{4} + πn, x = \frac{5 π}{6} + πn

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Frequently Asked Questions (FAQ)

What is the general solution for (cot(x)+1)(sqrt(3)tan(x)+1)=0 ?
The general solution for (cot(x)+1)(sqrt(3)tan(x)+1)=0 is x=(3pi)/4+pin,x=(5pi)/6+pin