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

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

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sqrt(3)csc(pi/2 x)=2

Solution

3 csc (\frac{π}{2} x) = 2

Solution

x = \frac{2}{3} + 4 n, x = \frac{4}{3} + 4 n

Degrees

x = 38.19718 \dots^{\circ} + 229.18311 \dots^{\circ} n, x = 76.39437 \dots^{\circ} + 229.18311 \dots^{\circ} n

Solution steps

3 csc (\frac{π}{2} x) = 2

Divide both sides by

3

3 csc (\frac{π}{2} x) = 2

Divide both sides by

3

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

Simplify

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

Simplify

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

\frac{3 csc ( \frac{π}{2} x )}{3}

Cancel the common factor:

3

= csc (\frac{π}{2} x)

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{π}{2} x) = \frac{2 3}{3}

csc (\frac{π}{2} x) = \frac{2 3}{3}

csc (\frac{π}{2} x) = \frac{2 3}{3}

General solutions for

csc (\frac{π}{2} 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

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

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

Solve

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

\frac{π}{2} x = \frac{π}{3} + 2 πn

Multiply both sides by

2

\frac{π}{2} x = \frac{π}{3} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

2 \cdot \frac{π}{2} x : π x

2 \cdot \frac{π}{2} x

Multiply fractions:

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

= \frac{2 π}{2} x

Cancel the common factor:

2

= x π

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

π x = \frac{2 π}{3} + 4 πn

π x = \frac{2 π}{3} + 4 πn

π x = \frac{2 π}{3} + 4 πn

Divide both sides by

π

π x = \frac{2 π}{3} + 4 πn

Divide both sides by

π

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

Simplify

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

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

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

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

\frac{\frac{2 π}{3}}{π} = \frac{2}{3}

\frac{\frac{2 π}{3}}{π}

Apply the fraction rule:

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

= \frac{2 π}{3 π}

Cancel the common factor:

π

= \frac{2}{3}

\frac{4 πn}{π} = 4 n

\frac{4 πn}{π}

Cancel the common factor:

π

= 4 n

= \frac{2}{3} + 4 n

x = \frac{2}{3} + 4 n

x = \frac{2}{3} + 4 n

x = \frac{2}{3} + 4 n

Solve

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

\frac{π}{2} x = \frac{2 π}{3} + 2 πn

Multiply both sides by

2

\frac{π}{2} x = \frac{2 π}{3} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

2 \cdot \frac{π}{2} x : π x

2 \cdot \frac{π}{2} x

Multiply fractions:

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

= \frac{2 π}{2} x

Cancel the common factor:

2

= x π

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

π x = \frac{4 π}{3} + 4 πn

π x = \frac{4 π}{3} + 4 πn

π x = \frac{4 π}{3} + 4 πn

Divide both sides by

π

π x = \frac{4 π}{3} + 4 πn

Divide both sides by

π

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

Simplify

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

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{\frac{4 π}{3}}{π} + \frac{4 πn}{π} : \frac{4}{3} + 4 n

\frac{\frac{4 π}{3}}{π} + \frac{4 πn}{π}

\frac{\frac{4 π}{3}}{π} = \frac{4}{3}

\frac{\frac{4 π}{3}}{π}

Apply the fraction rule:

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

= \frac{4 π}{3 π}

Cancel the common factor:

π

= \frac{4}{3}

\frac{4 πn}{π} = 4 n

\frac{4 πn}{π}

Cancel the common factor:

π

= 4 n

= \frac{4}{3} + 4 n

x = \frac{4}{3} + 4 n

x = \frac{4}{3} + 4 n

x = \frac{4}{3} + 4 n

x = \frac{2}{3} + 4 n, x = \frac{4}{3} + 4 n

Graph

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Frequently Asked Questions (FAQ)

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