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

The general solution for sin(pi/2 x)+1=0 is x=3+4n

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sin(pi/2 x)+1=0

Solution

sin (\frac{π}{2} x) + 1 = 0

Solution

x = 3 + 4 n

Degrees

x = 171.88733 \dots^{\circ} + 229.18311 \dots^{\circ} n

Solution steps

sin (\frac{π}{2} x) + 1 = 0

Move

1

to the right side

sin (\frac{π}{2} x) + 1 = 0

Subtract

1

from both sides

sin (\frac{π}{2} x) + 1 - 1 = 0 - 1

Simplify

sin (\frac{π}{2} x) = - 1

sin (\frac{π}{2} x) = - 1

General solutions for

sin (\frac{π}{2} x) = - 1

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}{2} x = \frac{3 π}{2} + 2 πn

\frac{π}{2} x = \frac{3 π}{2} + 2 πn

Solve

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

\frac{π}{2} x = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

\frac{π}{2} x = \frac{3 π}{2} + 2 πn

Multiply both sides by

2

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

Simplify

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

2 \cdot \frac{3 π}{2} + 2 \cdot 2 πn

2 \cdot \frac{3 π}{2} = 3 π

2 \cdot \frac{3 π}{2}

Multiply fractions:

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

= \frac{3 π 2}{2}

Cancel the common factor:

2

= 3 π

2 \cdot 2 πn = 4 πn

2 \cdot 2 πn

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn

= 3 π + 4 πn

π x = 3 π + 4 πn

π x = 3 π + 4 πn

π x = 3 π + 4 πn

Divide both sides by

π

π x = 3 π + 4 πn

Divide both sides by

π

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

Simplify

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

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

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

\frac{3 π}{π} + \frac{4 πn}{π}

Cancel

\frac{3 π}{π} : 3

\frac{3 π}{π}

Cancel the common factor:

π

= 3

= 3 + \frac{4 πn}{π}

Cancel

\frac{4 πn}{π} : 4 n

\frac{4 πn}{π}

Cancel the common factor:

π

= 4 n

= 3 + 4 n

x = 3 + 4 n

x = 3 + 4 n

x = 3 + 4 n

x = 3 + 4 n

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Frequently Asked Questions (FAQ)

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