What is the general solution for sin(3x-pi/2)=-(sqrt(3))/2 ,(3pi,4pi) ?

The general solution for sin(3x-pi/2)=-(sqrt(3))/2 ,(3pi,4pi) is x=(59pi)/(18),x=(61pi)/(18)

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sin(3x-pi/2)=-(sqrt(3))/2 ,(3pi,4pi)

Solution

sin (3 x - \frac{π}{2}) = - \frac{3}{2}, (3 π, 4 π)

Solution

x = \frac{59 π}{18}, x = \frac{61 π}{18}

Degrees

x = 59 0^{\circ}, x = 61 0^{\circ}

Solution steps

sin (3 x - \frac{π}{2}) = - \frac{3}{2}, (3 π, 4 π)

General solutions for

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

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}

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

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

Solve

3 x - \frac{π}{2} = \frac{4 π}{3} + 2 πn : x = \frac{2 πn}{3} + \frac{11 π}{18}

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

Move

\frac{π}{2}

to the right side

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

Add

\frac{π}{2}

to both sides

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

Simplify

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

Simplify

3 x - \frac{π}{2} + \frac{π}{2} : 3 x

3 x - \frac{π}{2} + \frac{π}{2}

Add similar elements:

- \frac{π}{2} + \frac{π}{2} = 0

= 3 x

Simplify

\frac{4 π}{3} + 2 πn + \frac{π}{2} : 2 πn + \frac{11 π}{6}

\frac{4 π}{3} + 2 πn + \frac{π}{2}

Group like terms

= 2 πn + \frac{π}{2} + \frac{4 π}{3}

Least Common Multiplier of

2, 3 : 6

2, 3

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Multiply each factor the greatest number of times it occurs in either

2

3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

6

For

\frac{π}{2} :

multiply the denominator and numerator by

3

\frac{π}{2} = \frac{π 3}{2 \cdot 3} = \frac{π 3}{6}

For

\frac{4 π}{3} :

multiply the denominator and numerator by

2

\frac{4 π}{3} = \frac{4 π 2}{3 \cdot 2} = \frac{8 π}{6}

= \frac{π 3}{6} + \frac{8 π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{π 3 + 8 π}{6}

Add similar elements:

3 π + 8 π = 11 π

= 2 πn + \frac{11 π}{6}

3 x = 2 πn + \frac{11 π}{6}

3 x = 2 πn + \frac{11 π}{6}

3 x = 2 πn + \frac{11 π}{6}

Divide both sides by

3

3 x = 2 πn + \frac{11 π}{6}

Divide both sides by

3

\frac{3 x}{3} = \frac{2 πn}{3} + \frac{\frac{11 π}{6}}{3}

Simplify

\frac{3 x}{3} = \frac{2 πn}{3} + \frac{\frac{11 π}{6}}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{2 πn}{3} + \frac{\frac{11 π}{6}}{3} : \frac{2 πn}{3} + \frac{11 π}{18}

\frac{2 πn}{3} + \frac{\frac{11 π}{6}}{3}

\frac{\frac{11 π}{6}}{3} = \frac{11 π}{18}

\frac{\frac{11 π}{6}}{3}

Apply the fraction rule:

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

= \frac{11 π}{6 \cdot 3}

Multiply the numbers:

6 \cdot 3 = 18

= \frac{11 π}{18}

= \frac{2 πn}{3} + \frac{11 π}{18}

x = \frac{2 πn}{3} + \frac{11 π}{18}

x = \frac{2 πn}{3} + \frac{11 π}{18}

x = \frac{2 πn}{3} + \frac{11 π}{18}

Solve

3 x - \frac{π}{2} = \frac{5 π}{3} + 2 πn : x = \frac{2 πn}{3} + \frac{13 π}{18}

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

Move

\frac{π}{2}

to the right side

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

Add

\frac{π}{2}

to both sides

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

Simplify

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

Simplify

3 x - \frac{π}{2} + \frac{π}{2} : 3 x

3 x - \frac{π}{2} + \frac{π}{2}

Add similar elements:

- \frac{π}{2} + \frac{π}{2} = 0

= 3 x

Simplify

\frac{5 π}{3} + 2 πn + \frac{π}{2} : 2 πn + \frac{13 π}{6}

\frac{5 π}{3} + 2 πn + \frac{π}{2}

Group like terms

= 2 πn + \frac{π}{2} + \frac{5 π}{3}

Least Common Multiplier of

2, 3 : 6

2, 3

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Multiply each factor the greatest number of times it occurs in either

2

3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

6

For

\frac{π}{2} :

multiply the denominator and numerator by

3

\frac{π}{2} = \frac{π 3}{2 \cdot 3} = \frac{π 3}{6}

For

\frac{5 π}{3} :

multiply the denominator and numerator by

2

\frac{5 π}{3} = \frac{5 π 2}{3 \cdot 2} = \frac{10 π}{6}

= \frac{π 3}{6} + \frac{10 π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{π 3 + 10 π}{6}

Add similar elements:

3 π + 10 π = 13 π

= 2 πn + \frac{13 π}{6}

3 x = 2 πn + \frac{13 π}{6}

3 x = 2 πn + \frac{13 π}{6}

3 x = 2 πn + \frac{13 π}{6}

Divide both sides by

3

3 x = 2 πn + \frac{13 π}{6}

Divide both sides by

3

\frac{3 x}{3} = \frac{2 πn}{3} + \frac{\frac{13 π}{6}}{3}

Simplify

\frac{3 x}{3} = \frac{2 πn}{3} + \frac{\frac{13 π}{6}}{3}

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{2 πn}{3} + \frac{\frac{13 π}{6}}{3} : \frac{2 πn}{3} + \frac{13 π}{18}

\frac{2 πn}{3} + \frac{\frac{13 π}{6}}{3}

\frac{\frac{13 π}{6}}{3} = \frac{13 π}{18}

\frac{\frac{13 π}{6}}{3}

Apply the fraction rule:

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

= \frac{13 π}{6 \cdot 3}

Multiply the numbers:

6 \cdot 3 = 18

= \frac{13 π}{18}

= \frac{2 πn}{3} + \frac{13 π}{18}

x = \frac{2 πn}{3} + \frac{13 π}{18}

x = \frac{2 πn}{3} + \frac{13 π}{18}

x = \frac{2 πn}{3} + \frac{13 π}{18}

x = \frac{2 πn}{3} + \frac{11 π}{18}, x = \frac{2 πn}{3} + \frac{13 π}{18}

Solutions for the range

(3 π, 4 π)

x = \frac{59 π}{18}, x = \frac{61 π}{18}

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Frequently Asked Questions (FAQ)

What is the general solution for sin(3x-pi/2)=-(sqrt(3))/2 ,(3pi,4pi) ?
The general solution for sin(3x-pi/2)=-(sqrt(3))/2 ,(3pi,4pi) is x=(59pi)/(18),x=(61pi)/(18)