What is the general solution for -5=-5+4sin(-2x+(3pi)/4) ?

The general solution for -5=-5+4sin(-2x+(3pi)/4) is x=-pin+(3pi)/8 ,x=-pin-pi/2+(3pi)/8

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-5=-5+4sin(-2x+(3pi)/4)

Solution

- 5 = - 5 + 4 sin (- 2 x + \frac{3 π}{4})

Solution

x = - πn + \frac{3 π}{8}, x = - πn - \frac{π}{2} + \frac{3 π}{8}

Degrees

x = 67. 5^{\circ} - 18 0^{\circ} n, x = - 22. 5^{\circ} - 18 0^{\circ} n

Solution steps

- 5 = - 5 + 4 sin (- 2 x + \frac{3 π}{4})

Switch sides

- 5 + 4 sin (- 2 x + \frac{3 π}{4}) = - 5

Add

5

to both sides

- 5 + 4 sin (- 2 x + \frac{3 π}{4}) + 5 = - 5 + 5

Simplify

4 sin (- 2 x + \frac{3 π}{4}) = 0

Divide both sides by

4

4 sin (- 2 x + \frac{3 π}{4}) = 0

Divide both sides by

4

\frac{4 sin ( - 2 x + \frac{3 π}{4} )}{4} = \frac{0}{4}

Simplify

sin (- 2 x + \frac{3 π}{4}) = 0

sin (- 2 x + \frac{3 π}{4}) = 0

General solutions for

sin (- 2 x + \frac{3 π}{4}) = 0

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}{4} = 0 + 2 πn, - 2 x + \frac{3 π}{4} = π + 2 πn

- 2 x + \frac{3 π}{4} = 0 + 2 πn, - 2 x + \frac{3 π}{4} = π + 2 πn

Solve

- 2 x + \frac{3 π}{4} = 0 + 2 πn : x = - πn + \frac{3 π}{8}

- 2 x + \frac{3 π}{4} = 0 + 2 πn

0 + 2 πn = 2 πn

- 2 x + \frac{3 π}{4} = 2 πn

Move

\frac{3 π}{4}

to the right side

- 2 x + \frac{3 π}{4} = 2 πn

Subtract

\frac{3 π}{4}

from both sides

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

Simplify

- 2 x = 2 πn - \frac{3 π}{4}

- 2 x = 2 πn - \frac{3 π}{4}

Divide both sides by

- 2

- 2 x = 2 πn - \frac{3 π}{4}

Divide both sides by

- 2

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

Simplify

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

Simplify

\frac{- 2 x}{- 2} : x

\frac{- 2 x}{- 2}

Apply the fraction rule:

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

= \frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{2 πn}{- 2} - \frac{\frac{3 π}{4}}{- 2} : - πn + \frac{3 π}{8}

\frac{2 πn}{- 2} - \frac{\frac{3 π}{4}}{- 2}

\frac{2 πn}{- 2} = - πn

\frac{2 πn}{- 2}

Apply the fraction rule:

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

= - \frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= - πn

= - πn - \frac{\frac{3 π}{4}}{- 2}

\frac{\frac{3 π}{4}}{- 2} = - \frac{3 π}{8}

\frac{\frac{3 π}{4}}{- 2}

Apply the fraction rule:

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

= - \frac{\frac{3 π}{4}}{2}

Apply the fraction rule:

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

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

= - \frac{3 π}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= - \frac{3 π}{8}

= - πn - (- \frac{3 π}{8})

Apply rule

- (- a) = a

= - πn + \frac{3 π}{8}

x = - πn + \frac{3 π}{8}

x = - πn + \frac{3 π}{8}

x = - πn + \frac{3 π}{8}

Solve

- 2 x + \frac{3 π}{4} = π + 2 πn : x = - πn - \frac{π}{2} + \frac{3 π}{8}

- 2 x + \frac{3 π}{4} = π + 2 πn

Move

\frac{3 π}{4}

to the right side

- 2 x + \frac{3 π}{4} = π + 2 πn

Subtract

\frac{3 π}{4}

from both sides

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

Simplify

- 2 x = π + 2 πn - \frac{3 π}{4}

- 2 x = π + 2 πn - \frac{3 π}{4}

Divide both sides by

- 2

- 2 x = π + 2 πn - \frac{3 π}{4}

Divide both sides by

- 2

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

Simplify

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

Simplify

\frac{- 2 x}{- 2} : x

\frac{- 2 x}{- 2}

Apply the fraction rule:

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

= \frac{2 x}{2}

Divide the numbers:

\frac{2}{2} = 1

= x

Simplify

\frac{π}{- 2} + \frac{2 πn}{- 2} - \frac{\frac{3 π}{4}}{- 2} : - πn - \frac{π}{2} + \frac{3 π}{8}

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

Apply the fraction rule:

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

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

\frac{2 πn}{- 2} = - πn

\frac{2 πn}{- 2}

Apply the fraction rule:

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

= - \frac{2 πn}{2}

Divide the numbers:

\frac{2}{2} = 1

= - πn

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

Group like terms

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

\frac{\frac{3 π}{4}}{- 2} = - \frac{3 π}{8}

\frac{\frac{3 π}{4}}{- 2}

Apply the fraction rule:

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

= - \frac{\frac{3 π}{4}}{2}

Apply the fraction rule:

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

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

= - \frac{3 π}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= - \frac{3 π}{8}

= - πn - \frac{π}{2} - (- \frac{3 π}{8})

Apply rule

- (- a) = a

= - πn - \frac{π}{2} + \frac{3 π}{8}

x = - πn - \frac{π}{2} + \frac{3 π}{8}

x = - πn - \frac{π}{2} + \frac{3 π}{8}

x = - πn - \frac{π}{2} + \frac{3 π}{8}

x = - πn + \frac{3 π}{8}, x = - πn - \frac{π}{2} + \frac{3 π}{8}

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Frequently Asked Questions (FAQ)

What is the general solution for -5=-5+4sin(-2x+(3pi)/4) ?
The general solution for -5=-5+4sin(-2x+(3pi)/4) is x=-pin+(3pi)/8 ,x=-pin-pi/2+(3pi)/8