What is the general solution for 4csc(x)=sqrt(3)csc(x)cot(x)+3csc(x) ?

The general solution for 4csc(x)=sqrt(3)csc(x)cot(x)+3csc(x) is x= pi/3+pin

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4csc(x)=sqrt(3)csc(x)cot(x)+3csc(x)

Solution

4 csc (x) = 3 csc (x) cot (x) + 3 csc (x)

Solution

x = \frac{π}{3} + πn

Degrees

x = 6 0^{\circ} + 18 0^{\circ} n

Solution steps

4 csc (x) = 3 csc (x) cot (x) + 3 csc (x)

Subtract

3 csc (x) cot (x) + 3 csc (x)

from both sides

csc (x) - 3 csc (x) cot (x) = 0

Factor

csc (x) - 3 csc (x) cot (x) : csc (x) (1 - 3 cot (x))

csc (x) - 3 csc (x) cot (x)

Factor out common term

csc (x)

= csc (x) (1 - 3 cot (x))

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

Solving each part separately

csc (x) = 0 or 1 - 3 cot (x) = 0

csc (x) = 0 :

No Solution

csc (x) = 0

csc (x) \leq - 1 or csc (x) \geq 1

No Solution

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

1 - 3 cot (x) = 0

Move

1

to the right side

1 - 3 cot (x) = 0

Subtract

1

from both sides

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

Simplify

- 3 cot (x) = - 1

- 3 cot (x) = - 1

Divide both sides by

- 3

- 3 cot (x) = - 1

Divide both sides by

- 3

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

Simplify

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

Simplify

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

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

Apply the fraction rule:

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

= \frac{3 cot ( x )}{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{π}{3} + πn

x = \frac{π}{3} + πn

Combine all the solutions

x = \frac{π}{3} + πn

Graph

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sin^{2} (x) = sin^{2} (\frac{x}{2})

cos(x)= 6/8

cos (x) = \frac{6}{8}

Frequently Asked Questions (FAQ)

What is the general solution for 4csc(x)=sqrt(3)csc(x)cot(x)+3csc(x) ?
The general solution for 4csc(x)=sqrt(3)csc(x)cot(x)+3csc(x) is x= pi/3+pin