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

The general solution for 2sin((pix)/2)+1=0 is x= 7/3+4n,x= 11/3+4n

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2sin((pix)/2)+1=0

Solution

2 sin (\frac{π x}{2}) + 1 = 0

Solution

x = \frac{7}{3} + 4 n, x = \frac{11}{3} + 4 n

Degrees

x = 133.69015 \dots^{\circ} + 229.18311 \dots^{\circ} n, x = 210.08452 \dots^{\circ} + 229.18311 \dots^{\circ} n

Solution steps

2 sin (\frac{π x}{2}) + 1 = 0

Move

1

to the right side

2 sin (\frac{π x}{2}) + 1 = 0

Subtract

1

from both sides

2 sin (\frac{π x}{2}) + 1 - 1 = 0 - 1

Simplify

2 sin (\frac{π x}{2}) = - 1

2 sin (\frac{π x}{2}) = - 1

Divide both sides by

2

2 sin (\frac{π x}{2}) = - 1

Divide both sides by

2

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

Simplify

sin (\frac{π x}{2}) = - \frac{1}{2}

sin (\frac{π x}{2}) = - \frac{1}{2}

General solutions for

sin (\frac{π x}{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}

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

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

Solve

\frac{π x}{2} = \frac{7 π}{6} + 2 πn : x = \frac{7}{3} + 4 n

\frac{π x}{2} = \frac{7 π}{6} + 2 πn

Multiply both sides by

2

\frac{π x}{2} = \frac{7 π}{6} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 π x}{2} : π x

\frac{2 π x}{2}

Divide the numbers:

\frac{2}{2} = 1

= π x

Simplify

2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn : \frac{7 π}{3} + 4 πn

2 \cdot \frac{7 π}{6} + 2 \cdot 2 πn

2 \cdot \frac{7 π}{6} = \frac{7 π}{3}

2 \cdot \frac{7 π}{6}

Multiply fractions:

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

= \frac{7 π 2}{6}

Multiply the numbers:

7 \cdot 2 = 14

= \frac{14 π}{6}

Cancel the common factor:

2

= \frac{7 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{7 π}{3} + 4 πn

π x = \frac{7 π}{3} + 4 πn

π x = \frac{7 π}{3} + 4 πn

π x = \frac{7 π}{3} + 4 πn

Divide both sides by

π

π x = \frac{7 π}{3} + 4 πn

Divide both sides by

π

\frac{π x}{π} = \frac{\frac{7 π}{3}}{π} + \frac{4 πn}{π}

Simplify

\frac{π x}{π} = \frac{\frac{7 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{7 π}{3}}{π} + \frac{4 πn}{π} : \frac{7}{3} + 4 n

\frac{\frac{7 π}{3}}{π} + \frac{4 πn}{π}

\frac{\frac{7 π}{3}}{π} = \frac{7}{3}

\frac{\frac{7 π}{3}}{π}

Apply the fraction rule:

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

= \frac{7 π}{3 π}

Cancel the common factor:

π

= \frac{7}{3}

\frac{4 πn}{π} = 4 n

\frac{4 πn}{π}

Cancel the common factor:

π

= 4 n

= \frac{7}{3} + 4 n

x = \frac{7}{3} + 4 n

x = \frac{7}{3} + 4 n

x = \frac{7}{3} + 4 n

Solve

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

\frac{π x}{2} = \frac{11 π}{6} + 2 πn

Multiply both sides by

2

\frac{π x}{2} = \frac{11 π}{6} + 2 πn

Multiply both sides by

2

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

Simplify

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

Simplify

\frac{2 π x}{2} : π x

\frac{2 π x}{2}

Divide the numbers:

\frac{2}{2} = 1

= π x

Simplify

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

2 \cdot \frac{11 π}{6} + 2 \cdot 2 πn

2 \cdot \frac{11 π}{6} = \frac{11 π}{3}

2 \cdot \frac{11 π}{6}

Multiply fractions:

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

= \frac{11 π 2}{6}

Multiply the numbers:

11 \cdot 2 = 22

= \frac{22 π}{6}

Cancel the common factor:

2

= \frac{11 π}{3}

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= \frac{11 π}{3} + 4 πn

π x = \frac{11 π}{3} + 4 πn

π x = \frac{11 π}{3} + 4 πn

π x = \frac{11 π}{3} + 4 πn

Divide both sides by

π

π x = \frac{11 π}{3} + 4 πn

Divide both sides by

π

\frac{π x}{π} = \frac{\frac{11 π}{3}}{π} + \frac{4 πn}{π}

Simplify

\frac{π x}{π} = \frac{\frac{11 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{11 π}{3}}{π} + \frac{4 πn}{π} : \frac{11}{3} + 4 n

\frac{\frac{11 π}{3}}{π} + \frac{4 πn}{π}

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

\frac{\frac{11 π}{3}}{π}

Apply the fraction rule:

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

= \frac{11 π}{3 π}

Cancel the common factor:

π

= \frac{11}{3}

\frac{4 πn}{π} = 4 n

\frac{4 πn}{π}

Cancel the common factor:

π

= 4 n

= \frac{11}{3} + 4 n

x = \frac{11}{3} + 4 n

x = \frac{11}{3} + 4 n

x = \frac{11}{3} + 4 n

x = \frac{7}{3} + 4 n, x = \frac{11}{3} + 4 n

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Frequently Asked Questions (FAQ)

What is the general solution for 2sin((pix)/2)+1=0 ?
The general solution for 2sin((pix)/2)+1=0 is x= 7/3+4n,x= 11/3+4n