What is the general solution for 2sin(2x+(3pi)/2)+1=0 ?

The general solution for 2sin(2x+(3pi)/2)+1=0 is x=pin-pi/6 ,x=pin+pi/6

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2sin(2x+(3pi)/2)+1=0

Solution

2 sin (2 x + \frac{3 π}{2}) + 1 = 0

Solution

x = πn - \frac{π}{6}, x = πn + \frac{π}{6}

Degrees

x = - 3 0^{\circ} + 18 0^{\circ} n, x = 3 0^{\circ} + 18 0^{\circ} n

Solution steps

2 sin (2 x + \frac{3 π}{2}) + 1 = 0

Move

1

to the right side

2 sin (2 x + \frac{3 π}{2}) + 1 = 0

Subtract

1

from both sides

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

Simplify

2 sin (2 x + \frac{3 π}{2}) = - 1

2 sin (2 x + \frac{3 π}{2}) = - 1

Divide both sides by

2

2 sin (2 x + \frac{3 π}{2}) = - 1

Divide both sides by

2

\frac{2 sin ( 2 x + \frac{3 π}{2} )}{2} = \frac{- 1}{2}

Simplify

sin (2 x + \frac{3 π}{2}) = - \frac{1}{2}

sin (2 x + \frac{3 π}{2}) = - \frac{1}{2}

General solutions for

sin (2 x + \frac{3 π}{2}) = - \frac{1}{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}

2 x + \frac{3 π}{2} = \frac{7 π}{6} + 2 πn, 2 x + \frac{3 π}{2} = \frac{11 π}{6} + 2 πn

2 x + \frac{3 π}{2} = \frac{7 π}{6} + 2 πn, 2 x + \frac{3 π}{2} = \frac{11 π}{6} + 2 πn

Solve

2 x + \frac{3 π}{2} = \frac{7 π}{6} + 2 πn : x = πn - \frac{π}{6}

2 x + \frac{3 π}{2} = \frac{7 π}{6} + 2 πn

Move

\frac{3 π}{2}

to the right side

2 x + \frac{3 π}{2} = \frac{7 π}{6} + 2 πn

Subtract

\frac{3 π}{2}

from both sides

2 x + \frac{3 π}{2} - \frac{3 π}{2} = \frac{7 π}{6} + 2 πn - \frac{3 π}{2}

Simplify

2 x + \frac{3 π}{2} - \frac{3 π}{2} = \frac{7 π}{6} + 2 πn - \frac{3 π}{2}

Simplify

2 x + \frac{3 π}{2} - \frac{3 π}{2} : 2 x

2 x + \frac{3 π}{2} - \frac{3 π}{2}

Add similar elements:

\frac{3 π}{2} - \frac{3 π}{2} = 0

= 2 x

Simplify

\frac{7 π}{6} + 2 πn - \frac{3 π}{2} : 2 πn - \frac{π}{3}

\frac{7 π}{6} + 2 πn - \frac{3 π}{2}

Group like terms

= 2 πn + \frac{7 π}{6} - \frac{3 π}{2}

Least Common Multiplier of

6, 2 : 6

6, 2

Least Common Multiplier (LCM)

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

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

6

2

= 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{3 π}{2} :

multiply the denominator and numerator by

3

\frac{3 π}{2} = \frac{3 π 3}{2 \cdot 3} = \frac{9 π}{6}

= \frac{7 π}{6} - \frac{9 π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{7 π - 9 π}{6}

Add similar elements:

7 π - 9 π = - 2 π

= \frac{- 2 π}{6}

Apply the fraction rule:

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

= - \frac{2 π}{6}

Cancel the common factor:

2

= 2 πn - \frac{π}{3}

2 x = 2 πn - \frac{π}{3}

2 x = 2 πn - \frac{π}{3}

2 x = 2 πn - \frac{π}{3}

Divide both sides by

2

2 x = 2 πn - \frac{π}{3}

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{2 πn}{2} - \frac{\frac{π}{3}}{2} : πn - \frac{π}{6}

\frac{2 πn}{2} - \frac{\frac{π}{3}}{2}

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

\frac{\frac{π}{3}}{2} = \frac{π}{6}

\frac{\frac{π}{3}}{2}

Apply the fraction rule:

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

= \frac{π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{π}{6}

= πn - \frac{π}{6}

x = πn - \frac{π}{6}

x = πn - \frac{π}{6}

x = πn - \frac{π}{6}

Solve

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

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

Move

\frac{3 π}{2}

to the right side

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

Subtract

\frac{3 π}{2}

from both sides

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

Simplify

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

Simplify

2 x + \frac{3 π}{2} - \frac{3 π}{2} : 2 x

2 x + \frac{3 π}{2} - \frac{3 π}{2}

Add similar elements:

\frac{3 π}{2} - \frac{3 π}{2} = 0

= 2 x

Simplify

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

\frac{11 π}{6} + 2 πn - \frac{3 π}{2}

Group like terms

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

Least Common Multiplier of

2, 6 : 6

2, 6

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

6 : 2 \cdot 3

6

6

divides by

26 = 3 \cdot 2

= 2 \cdot 3

2, 3

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 3

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

2

6

= 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{3 π}{2} :

multiply the denominator and numerator by

3

\frac{3 π}{2} = \frac{3 π 3}{2 \cdot 3} = \frac{9 π}{6}

= - \frac{9 π}{6} + \frac{11 π}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{- 9 π + 11 π}{6}

Add similar elements:

- 9 π + 11 π = 2 π

= \frac{2 π}{6}

Cancel the common factor:

2

= 2 πn + \frac{π}{3}

2 x = 2 πn + \frac{π}{3}

2 x = 2 πn + \frac{π}{3}

2 x = 2 πn + \frac{π}{3}

Divide both sides by

2

2 x = 2 πn + \frac{π}{3}

Divide both sides by

2

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

Simplify

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

Simplify

\frac{2 x}{2} : x

\frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

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

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

\frac{2 πn}{2} = πn

\frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= πn

\frac{\frac{π}{3}}{2} = \frac{π}{6}

\frac{\frac{π}{3}}{2}

Apply the fraction rule:

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

= \frac{π}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{π}{6}

= πn + \frac{π}{6}

x = πn + \frac{π}{6}

x = πn + \frac{π}{6}

x = πn + \frac{π}{6}

x = πn - \frac{π}{6}, x = πn + \frac{π}{6}

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Frequently Asked Questions (FAQ)

What is the general solution for 2sin(2x+(3pi)/2)+1=0 ?
The general solution for 2sin(2x+(3pi)/2)+1=0 is x=pin-pi/6 ,x=pin+pi/6