What is the general solution for 8|sin(pix)-1/4 |+1=7 ?

The general solution for 8|sin(pix)-1/4 |+1=7 is x= 7/6+2n,x= 11/6+2n,x= 1/2+2n

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8|sin(pix)-1/4 |+1=7

Solution

8 sin (π x) - \frac{1}{4} + 1 = 7

Solution

x = \frac{7}{6} + 2 n, x = \frac{11}{6} + 2 n, x = \frac{1}{2} + 2 n

Degrees

x = 66.84507 \dots^{\circ} + 114.59155 \dots^{\circ} n, x = 105.04226 \dots^{\circ} + 114.59155 \dots^{\circ} n, x = 28.64788 \dots^{\circ} + 114.59155 \dots^{\circ} n

Solution steps

8 sin (π x) - \frac{1}{4} + 1 = 7

Solve by substitution

8 sin (π x) - \frac{1}{4} + 1 = 7

Let:

sin (π x) = u

8 u - \frac{1}{4} + 1 = 7

8 u - \frac{1}{4} + 1 = 7 : u = - \frac{1}{2} or u = 1

8 u - \frac{1}{4} + 1 = 7

Move

1

to the right side

8 u - \frac{1}{4} + 1 = 7

Subtract

1

from both sides

8 u - \frac{1}{4} + 1 - 1 = 7 - 1

Simplify

8 u - \frac{1}{4} = 6

8 u - \frac{1}{4} = 6

Divide both sides by

8

8 u - \frac{1}{4} = 6

Divide both sides by

8

\frac{8 u - \frac{1}{4}}{8} = \frac{6}{8}

Simplify

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

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

Apply absolute rule: If

∣ u ∣ = a, a > 0

then

u = a

u = - a

u - \frac{1}{4} = - \frac{3}{4} or u - \frac{1}{4} = \frac{3}{4}

u - \frac{1}{4} = - \frac{3}{4} or u - \frac{1}{4} = \frac{3}{4}

u - \frac{1}{4} = - \frac{3}{4} or u - \frac{1}{4} = \frac{3}{4}

u - \frac{1}{4} = - \frac{3}{4} : u = - \frac{1}{2}

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

Move

\frac{1}{4}

to the right side

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

Add

\frac{1}{4}

to both sides

u - \frac{1}{4} + \frac{1}{4} = - \frac{3}{4} + \frac{1}{4}

Simplify

u - \frac{1}{4} + \frac{1}{4} = - \frac{3}{4} + \frac{1}{4}

Simplify

u - \frac{1}{4} + \frac{1}{4} : u

u - \frac{1}{4} + \frac{1}{4}

Add similar elements:

- \frac{1}{4} + \frac{1}{4} = 0

= u

Simplify

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

- \frac{3}{4} + \frac{1}{4}

Apply rule

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

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

Add/Subtract the numbers:

- 3 + 1 = - 2

= \frac{- 2}{4}

Apply the fraction rule:

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

= - \frac{2}{4}

Cancel the common factor:

2

= - \frac{1}{2}

u = - \frac{1}{2}

u = - \frac{1}{2}

u = - \frac{1}{2}

u - \frac{1}{4} = \frac{3}{4} : u = 1

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

Move

\frac{1}{4}

to the right side

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

Add

\frac{1}{4}

to both sides

u - \frac{1}{4} + \frac{1}{4} = \frac{3}{4} + \frac{1}{4}

Simplify

u - \frac{1}{4} + \frac{1}{4} = \frac{3}{4} + \frac{1}{4}

Simplify

u - \frac{1}{4} + \frac{1}{4} : u

u - \frac{1}{4} + \frac{1}{4}

Add similar elements:

- \frac{1}{4} + \frac{1}{4} = 0

= u

Simplify

\frac{3}{4} + \frac{1}{4} : 1

\frac{3}{4} + \frac{1}{4}

Apply rule

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

= \frac{3 + 1}{4}

Add the numbers:

3 + 1 = 4

= \frac{4}{4}

Apply rule

\frac{a}{a} = 1

= 1

u = 1

u = 1

u = 1

Combine Solutions:

u = - \frac{1}{2} or u = 1

u = - \frac{1}{2} or u = 1

Substitute back

u = sin (π x)

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

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

sin (π x) = - \frac{1}{2} : x = \frac{7}{6} + 2 n, x = \frac{11}{6} + 2 n

sin (π x) = - \frac{1}{2}

General solutions for

sin (π x) = - \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}

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

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

Solve

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

π x = \frac{7 π}{6} + 2 πn

Divide both sides by

π

π x = \frac{7 π}{6} + 2 πn

Divide both sides by

π

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

Simplify

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

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{\frac{7 π}{6}}{π} + \frac{2 πn}{π} : \frac{7}{6} + 2 n

\frac{\frac{7 π}{6}}{π} + \frac{2 πn}{π}

\frac{\frac{7 π}{6}}{π} = \frac{7}{6}

\frac{\frac{7 π}{6}}{π}

Apply the fraction rule:

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

= \frac{7 π}{6 π}

Cancel the common factor:

π

= \frac{7}{6}

\frac{2 πn}{π} = 2 n

\frac{2 πn}{π}

Cancel the common factor:

π

= 2 n

= \frac{7}{6} + 2 n

x = \frac{7}{6} + 2 n

x = \frac{7}{6} + 2 n

x = \frac{7}{6} + 2 n

Solve

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

π x = \frac{11 π}{6} + 2 πn

Divide both sides by

π

π x = \frac{11 π}{6} + 2 πn

Divide both sides by

π

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

Simplify

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

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{\frac{11 π}{6}}{π} + \frac{2 πn}{π} : \frac{11}{6} + 2 n

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

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

\frac{\frac{11 π}{6}}{π}

Apply the fraction rule:

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

= \frac{11 π}{6 π}

Cancel the common factor:

π

= \frac{11}{6}

\frac{2 πn}{π} = 2 n

\frac{2 πn}{π}

Cancel the common factor:

π

= 2 n

= \frac{11}{6} + 2 n

x = \frac{11}{6} + 2 n

x = \frac{11}{6} + 2 n

x = \frac{11}{6} + 2 n

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

sin (π x) = 1 : x = \frac{1}{2} + 2 n

sin (π x) = 1

General solutions for

sin (π 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}

π x = \frac{π}{2} + 2 πn

π x = \frac{π}{2} + 2 πn

Solve

π x = \frac{π}{2} + 2 πn : x = \frac{1}{2} + 2 n

π x = \frac{π}{2} + 2 πn

Divide both sides by

π

π x = \frac{π}{2} + 2 πn

Divide both sides by

π

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

Simplify

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

Simplify

\frac{π x}{π} : x

\frac{π x}{π}

Cancel the common factor:

π

= x

Simplify

\frac{\frac{π}{2}}{π} + \frac{2 πn}{π} : \frac{1}{2} + 2 n

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

\frac{\frac{π}{2}}{π} = \frac{1}{2}

\frac{\frac{π}{2}}{π}

Apply the fraction rule:

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

= \frac{π}{2 π}

Cancel the common factor:

π

= \frac{1}{2}

\frac{2 πn}{π} = 2 n

\frac{2 πn}{π}

Cancel the common factor:

π

= 2 n

= \frac{1}{2} + 2 n

x = \frac{1}{2} + 2 n

x = \frac{1}{2} + 2 n

x = \frac{1}{2} + 2 n

x = \frac{1}{2} + 2 n

Combine all the solutions

x = \frac{7}{6} + 2 n, x = \frac{11}{6} + 2 n, x = \frac{1}{2} + 2 n

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Frequently Asked Questions (FAQ)

What is the general solution for 8|sin(pix)-1/4 |+1=7 ?
The general solution for 8|sin(pix)-1/4 |+1=7 is x= 7/6+2n,x= 11/6+2n,x= 1/2+2n