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

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

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sin((3(pi-x))/2)=0

Solution

sin (\frac{3 ( π - x )}{2}) = 0

Solution

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

Degrees

x = 18 0^{\circ} - 24 0^{\circ} n, x = 6 0^{\circ} - 24 0^{\circ} n

Solution steps

sin (\frac{3 ( π - x )}{2}) = 0

General solutions for

sin (\frac{3 ( π - x )}{2}) = 0

sin (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} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 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} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

\frac{3 ( π - x )}{2} = 0 + 2 πn, \frac{3 ( π - x )}{2} = π + 2 πn

\frac{3 ( π - x )}{2} = 0 + 2 πn, \frac{3 ( π - x )}{2} = π + 2 πn

Solve

\frac{3 ( π - x )}{2} = 0 + 2 πn : x = - \frac{4 πn}{3} + π

\frac{3 ( π - x )}{2} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{3 ( π - x )}{2} = 2 πn

Multiply both sides by

2

\frac{3 ( π - x )}{2} = 2 πn

Multiply both sides by

2

\frac{2 \cdot 3 ( π - x )}{2} = 2 \cdot 2 πn

Simplify

3 (- x + π) = 4 πn

3 (- x + π) = 4 πn

Divide both sides by

3

3 (- x + π) = 4 πn

Divide both sides by

3

\frac{3 ( - x + π )}{3} = \frac{4 πn}{3}

Simplify

- x + π = \frac{4 πn}{3}

- x + π = \frac{4 πn}{3}

Move

π

to the right side

- x + π = \frac{4 πn}{3}

Subtract

π

from both sides

- x + π - π = \frac{4 πn}{3} - π

Simplify

- x = \frac{4 πn}{3} - π

- x = \frac{4 πn}{3} - π

Divide both sides by

- 1

- x = \frac{4 πn}{3} - π

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{\frac{4 πn}{3}}{- 1} - \frac{π}{- 1}

Simplify

\frac{- x}{- 1} = \frac{\frac{4 πn}{3}}{- 1} - \frac{π}{- 1}

Simplify

\frac{- x}{- 1} : x

\frac{- x}{- 1}

Apply the fraction rule:

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

= \frac{x}{1}

Apply rule

\frac{a}{1} = a

= x

Simplify

\frac{\frac{4 πn}{3}}{- 1} - \frac{π}{- 1} : - \frac{4 πn}{3} + π

\frac{\frac{4 πn}{3}}{- 1} - \frac{π}{- 1}

\frac{\frac{4 πn}{3}}{- 1} = - \frac{4 πn}{3}

\frac{\frac{4 πn}{3}}{- 1}

Apply the fraction rule:

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

= - \frac{\frac{4 πn}{3}}{1}

Apply the fraction rule:

\frac{a}{1} = a

\frac{\frac{4 πn}{3}}{1} = \frac{4 πn}{3}

= - \frac{4 πn}{3}

= - \frac{4 πn}{3} - \frac{π}{- 1}

\frac{π}{- 1} = - π

\frac{π}{- 1}

Apply the fraction rule:

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

= - \frac{π}{1}

Apply rule

\frac{a}{1} = a

= - π

= - \frac{4 πn}{3} - (- π)

Apply rule

- (- a) = a

= - \frac{4 πn}{3} + π

x = - \frac{4 πn}{3} + π

x = - \frac{4 πn}{3} + π

x = - \frac{4 πn}{3} + π

Solve

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

\frac{3 ( π - x )}{2} = π + 2 πn

Multiply both sides by

2

\frac{3 ( π - x )}{2} = π + 2 πn

Multiply both sides by

2

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

Simplify

3 (- x + π) = 2 π + 4 πn

3 (- x + π) = 2 π + 4 πn

Divide both sides by

3

3 (- x + π) = 2 π + 4 πn

Divide both sides by

3

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

Simplify

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

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

Move

π

to the right side

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

Subtract

π

from both sides

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

Simplify

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

Simplify

- x + π - π : - x

- x + π - π

Add similar elements:

π - π = 0

= - x

Simplify

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

\frac{2 π}{3} + \frac{4 πn}{3} - π

Combine the fractions

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

Apply rule

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

= \frac{2 π + 4 πn}{3}

= \frac{2 π + 4 πn}{3} - π

- x = \frac{2 π + 4 πn}{3} - π

- x = \frac{2 π + 4 πn}{3} - π

- x = \frac{2 π + 4 πn}{3} - π

Divide both sides by

- 1

- x = \frac{2 π + 4 πn}{3} - π

Divide both sides by

- 1

\frac{- x}{- 1} = \frac{\frac{2 π + 4 πn}{3}}{- 1} - \frac{π}{- 1}

Simplify

\frac{- x}{- 1} = \frac{\frac{2 π + 4 πn}{3}}{- 1} - \frac{π}{- 1}

Simplify

\frac{- x}{- 1} : x

\frac{- x}{- 1}

Apply the fraction rule:

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

= \frac{x}{1}

Apply rule

\frac{a}{1} = a

= x

Simplify

\frac{\frac{2 π + 4 πn}{3}}{- 1} - \frac{π}{- 1} : - \frac{2 π + 4 πn}{3} + π

\frac{\frac{2 π + 4 πn}{3}}{- 1} - \frac{π}{- 1}

\frac{\frac{2 π + 4 πn}{3}}{- 1} = - \frac{2 π + 4 πn}{3}

\frac{\frac{2 π + 4 πn}{3}}{- 1}

Apply the fraction rule:

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

= - \frac{\frac{2 π + 4 πn}{3}}{1}

Apply the fraction rule:

\frac{a}{1} = a

\frac{\frac{2 π + 4 πn}{3}}{1} = \frac{2 π + 4 πn}{3}

= - \frac{2 π + 4 πn}{3}

= - \frac{2 π + 4 πn}{3} - \frac{π}{- 1}

\frac{π}{- 1} = - π

\frac{π}{- 1}

Apply the fraction rule:

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

= - \frac{π}{1}

Apply rule

\frac{a}{1} = a

= - π

= - \frac{2 π + 4 πn}{3} - (- π)

Apply rule

- (- a) = a

= - \frac{2 π + 4 πn}{3} + π

x = - \frac{2 π + 4 πn}{3} + π

x = - \frac{2 π + 4 πn}{3} + π

x = - \frac{2 π + 4 πn}{3} + π

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

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Frequently Asked Questions (FAQ)

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