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

The general solution for sqrt(3)csc^2(x)+2csc(x)=0 is x=(4pi)/3+2pin,x=(5pi)/3+2pin

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sqrt(3)csc^2(x)+2csc(x)=0

Solution

3 csc^{2} (x) + 2 csc (x) = 0

Solution

x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Degrees

x = 24 0^{\circ} + 36 0^{\circ} n, x = 30 0^{\circ} + 36 0^{\circ} n

Solution steps

3 csc^{2} (x) + 2 csc (x) = 0

Solve by substitution

3 csc^{2} (x) + 2 csc (x) = 0

Let:

csc (x) = u

3 u^{2} + 2 u = 0

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

3 u^{2} + 2 u = 0

Solve with the quadratic formula

3 u^{2} + 2 u = 0

Quadratic Equation Formula:

For

a = 3, b = 2, c = 0

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 3 \cdot 0}{2 3}

u_{1, 2} = \frac{- 2 \pm 2 ^{2} - 4 3 \cdot 0}{2 3}

2^{2} - 43 \cdot 0 = 2

2^{2} - 43 \cdot 0

Apply rule

0 \cdot a = 0

= 2^{2} - 0

2^{2} - 0 = 2^{2}

= 2^{2}

Apply radical rule:

assuming

a \geq 0

= 2

u_{1, 2} = \frac{- 2 \pm 2}{2 3}

Separate the solutions

u_{1} = \frac{- 2 + 2}{2 3}, u_{2} = \frac{- 2 - 2}{2 3}

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

\frac{- 2 + 2}{2 3}

Add/Subtract the numbers:

- 2 + 2 = 0

= \frac{0}{2 3}

Apply rule

\frac{0}{a} = 0, a \neq = 0

= 0

u = \frac{- 2 - 2}{2 3} : - \frac{2 3}{3}

\frac{- 2 - 2}{2 3}

Subtract the numbers:

- 2 - 2 = - 4

= \frac{- 4}{2 3}

Apply the fraction rule:

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

= - \frac{4}{2 3}

Divide the numbers:

\frac{4}{2} = 2

= - \frac{2}{3}

Rationalize

- \frac{2}{3} : - \frac{2 3}{3}

- \frac{2}{3}

Multiply by the conjugate

\frac{3}{3}

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

33 = 3

33

Apply radical rule:

a a = a

33 = 3

= 3

= - \frac{2 3}{3}

= - \frac{2 3}{3}

The solutions to the quadratic equation are:

u = 0, u = - \frac{2 3}{3}

Substitute back

u = csc (x)

csc (x) = 0, csc (x) = - \frac{2 3}{3}

csc (x) = 0, csc (x) = - \frac{2 3}{3}

csc (x) = 0 :

No Solution

csc (x) = 0

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

No Solution

csc (x) = - \frac{2 3}{3} : x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

csc (x) = - \frac{2 3}{3}

General solutions for

csc (x) = - \frac{2 3}{3}

csc (x)

periodicity table with

2 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} csc (x) Undefiend 22 \frac{2 3}{3} 1 \frac{2 3}{3} 22 x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} csc (x) Undefiend - 2 - 2 - \frac{2 3}{3} - 1 - \frac{2 3}{3} - 2 - 2

x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Combine all the solutions

x = \frac{4 π}{3} + 2 πn, x = \frac{5 π}{3} + 2 πn

Popular Examples

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7 sin (2 x) sin (x) = 7 cos (x)

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csc (x) + cot (x) = 3

sin^2(x)-sin(x)+1=cos^2(x)

sin^{2} (x) - sin (x) + 1 = cos^{2} (x)

sin(x/2)=(sqrt(3))/2

sin (\frac{x}{2}) = \frac{3}{2}

cos(x)tan(x)-cos(x)=0

cos (x) tan (x) - cos (x) = 0

Frequently Asked Questions (FAQ)

What is the general solution for sqrt(3)csc^2(x)+2csc(x)=0 ?
The general solution for sqrt(3)csc^2(x)+2csc(x)=0 is x=(4pi)/3+2pin,x=(5pi)/3+2pin