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

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

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sqrt(3)csc(x/2)-2=0

Solution

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

Solution

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

Degrees

x = 12 0^{\circ} + 72 0^{\circ} n, x = 24 0^{\circ} + 72 0^{\circ} n

Solution steps

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

Move

2

to the right side

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

Add

2

to both sides

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

Simplify

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

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

Divide both sides by

3

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

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 csc ( \frac{x}{2} )}{3} : csc (\frac{x}{2})

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

Cancel the common factor:

3

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

Simplify

\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}

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

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

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

General solutions for

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

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

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

Solve

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

\frac{x}{2} = \frac{π}{3} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{π}{3} + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot \frac{π}{3} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} = 2 \cdot \frac{π}{3} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{π}{3} + 2 \cdot 2 πn : \frac{2 π}{3} + 4 πn

2 \cdot \frac{π}{3} + 2 \cdot 2 πn

Multiply

2 \cdot \frac{π}{3} : \frac{2 π}{3}

2 \cdot \frac{π}{3}

Multiply fractions:

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

= \frac{π 2}{3}

= \frac{2 π}{3} + 2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= \frac{2 π}{3} + 4 πn

x = \frac{2 π}{3} + 4 πn

x = \frac{2 π}{3} + 4 πn

x = \frac{2 π}{3} + 4 πn

Solve

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

\frac{x}{2} = \frac{2 π}{3} + 2 πn

Multiply both sides by

2

\frac{x}{2} = \frac{2 π}{3} + 2 πn

Multiply both sides by

2

\frac{2 x}{2} = 2 \cdot \frac{2 π}{3} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} = 2 \cdot \frac{2 π}{3} + 2 \cdot 2 πn

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

2 \cdot \frac{2 π}{3} + 2 \cdot 2 πn : \frac{4 π}{3} + 4 πn

2 \cdot \frac{2 π}{3} + 2 \cdot 2 πn

2 \cdot \frac{2 π}{3} = \frac{4 π}{3}

2 \cdot \frac{2 π}{3}

Multiply fractions:

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

= \frac{2 π 2}{3}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{4 π}{3} + 4 πn

x = \frac{4 π}{3} + 4 πn

x = \frac{4 π}{3} + 4 πn

x = \frac{4 π}{3} + 4 πn

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

Popular Examples

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arcsin (y) = \frac{1}{2}

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tan(θ)=-(sqrt(3))/3 ,0<= θ<= 2pi

tan (θ) = - \frac{3}{3}, 0 \leq θ \leq 2 π

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

sinh^2(x)-5cosh(x)+7=0

sinh^{2} (x) - 5 cosh (x) + 7 = 0

Frequently Asked Questions (FAQ)

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