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Popular Trigonometry >

solvefor x,y-1=-4+3sin(4x-pi/4)

Solution

solvefor

Solution

x = \frac{arcsin ( \frac{y + 3}{3} )}{4} + \frac{πn}{2} + \frac{π}{16}, x = \frac{π}{4} + \frac{π}{16} + \frac{πn}{2} + \frac{arcsin ( - \frac{y + 3}{3} )}{4}

Solution steps

y - 1 = - 4 + 3 sin (4 x - \frac{π}{4})

Switch sides

- 4 + 3 sin (4 x - \frac{π}{4}) = y - 1

Move

4

to the right side

- 4 + 3 sin (4 x - \frac{π}{4}) = y - 1

Add

4

to both sides

- 4 + 3 sin (4 x - \frac{π}{4}) + 4 = y - 1 + 4

Simplify

3 sin (4 x - \frac{π}{4}) = y + 3

3 sin (4 x - \frac{π}{4}) = y + 3

Divide both sides by

3

3 sin (4 x - \frac{π}{4}) = y + 3

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 sin ( 4 x - \frac{π}{4} )}{3} : sin (4 x - \frac{π}{4})

\frac{3 sin ( 4 x - \frac{π}{4} )}{3}

Divide the numbers:

\frac{3}{3} = 1

= sin (4 x - \frac{π}{4})

Simplify

\frac{y}{3} + \frac{3}{3} : \frac{y + 3}{3}

\frac{y}{3} + \frac{3}{3}

Apply rule

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

= \frac{y + 3}{3}

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

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

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

Apply trig inverse properties

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

General solutions for

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

sin (x) = a \Rightarrow x = arcsin (a) + 2 πn, x = π + arcsin (a) + 2 πn

4 x - \frac{π}{4} = arcsin (\frac{y + 3}{3}) + 2 πn, 4 x - \frac{π}{4} = π + arcsin (- \frac{y + 3}{3}) + 2 πn

4 x - \frac{π}{4} = arcsin (\frac{y + 3}{3}) + 2 πn, 4 x - \frac{π}{4} = π + arcsin (- \frac{y + 3}{3}) + 2 πn

Solve

4 x - \frac{π}{4} = arcsin (\frac{y + 3}{3}) + 2 πn : x = \frac{arcsin ( \frac{y + 3}{3} )}{4} + \frac{πn}{2} + \frac{π}{16}

4 x - \frac{π}{4} = arcsin (\frac{y + 3}{3}) + 2 πn

Move

\frac{π}{4}

to the right side

4 x - \frac{π}{4} = arcsin (\frac{y + 3}{3}) + 2 πn

Add

\frac{π}{4}

to both sides

4 x - \frac{π}{4} + \frac{π}{4} = arcsin (\frac{y + 3}{3}) + 2 πn + \frac{π}{4}

Simplify

4 x = arcsin (\frac{y + 3}{3}) + 2 πn + \frac{π}{4}

4 x = arcsin (\frac{y + 3}{3}) + 2 πn + \frac{π}{4}

Divide both sides by

4

4 x = arcsin (\frac{y + 3}{3}) + 2 πn + \frac{π}{4}

Divide both sides by

4

\frac{4 x}{4} = \frac{arcsin ( \frac{y + 3}{3} )}{4} + \frac{2 πn}{4} + \frac{\frac{π}{4}}{4}

Simplify

\frac{4 x}{4} = \frac{arcsin ( \frac{y + 3}{3} )}{4} + \frac{2 πn}{4} + \frac{\frac{π}{4}}{4}

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{arcsin ( \frac{y + 3}{3} )}{4} + \frac{2 πn}{4} + \frac{\frac{π}{4}}{4} : \frac{arcsin ( \frac{y + 3}{3} )}{4} + \frac{πn}{2} + \frac{π}{16}

\frac{arcsin ( \frac{y + 3}{3} )}{4} + \frac{2 πn}{4} + \frac{\frac{π}{4}}{4}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Cancel the common factor:

2

= \frac{πn}{2}

\frac{\frac{π}{4}}{4} = \frac{π}{16}

\frac{\frac{π}{4}}{4}

Apply the fraction rule:

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

= \frac{π}{4 \cdot 4}

Multiply the numbers:

4 \cdot 4 = 16

= \frac{π}{16}

= \frac{arcsin ( \frac{y + 3}{3} )}{4} + \frac{πn}{2} + \frac{π}{16}

x = \frac{arcsin ( \frac{y + 3}{3} )}{4} + \frac{πn}{2} + \frac{π}{16}

x = \frac{arcsin ( \frac{y + 3}{3} )}{4} + \frac{πn}{2} + \frac{π}{16}

x = \frac{arcsin ( \frac{y + 3}{3} )}{4} + \frac{πn}{2} + \frac{π}{16}

Solve

4 x - \frac{π}{4} = π + arcsin (- \frac{y + 3}{3}) + 2 πn : x = \frac{π}{4} + \frac{π}{16} + \frac{πn}{2} + \frac{arcsin ( - \frac{y + 3}{3} )}{4}

4 x - \frac{π}{4} = π + arcsin (- \frac{y + 3}{3}) + 2 πn

Move

\frac{π}{4}

to the right side

4 x - \frac{π}{4} = π + arcsin (- \frac{y + 3}{3}) + 2 πn

Add

\frac{π}{4}

to both sides

4 x - \frac{π}{4} + \frac{π}{4} = π + arcsin (- \frac{y + 3}{3}) + 2 πn + \frac{π}{4}

Simplify

4 x = π + arcsin (- \frac{y + 3}{3}) + 2 πn + \frac{π}{4}

4 x = π + arcsin (- \frac{y + 3}{3}) + 2 πn + \frac{π}{4}

Divide both sides by

4

4 x = π + arcsin (- \frac{y + 3}{3}) + 2 πn + \frac{π}{4}

Divide both sides by

4

\frac{4 x}{4} = \frac{π}{4} + \frac{arcsin ( - \frac{y + 3}{3} )}{4} + \frac{2 πn}{4} + \frac{\frac{π}{4}}{4}

Simplify

\frac{4 x}{4} = \frac{π}{4} + \frac{arcsin ( - \frac{y + 3}{3} )}{4} + \frac{2 πn}{4} + \frac{\frac{π}{4}}{4}

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{π}{4} + \frac{arcsin ( - \frac{y + 3}{3} )}{4} + \frac{2 πn}{4} + \frac{\frac{π}{4}}{4} : \frac{π}{4} + \frac{π}{16} + \frac{πn}{2} + \frac{arcsin ( - \frac{y + 3}{3} )}{4}

\frac{π}{4} + \frac{arcsin ( - \frac{y + 3}{3} )}{4} + \frac{2 πn}{4} + \frac{\frac{π}{4}}{4}

Group like terms

= \frac{π}{4} + \frac{2 πn}{4} + \frac{\frac{π}{4}}{4} + \frac{arcsin ( - \frac{y + 3}{3} )}{4}

\frac{2 πn}{4} = \frac{πn}{2}

\frac{2 πn}{4}

Cancel the common factor:

2

= \frac{πn}{2}

\frac{\frac{π}{4}}{4} = \frac{π}{16}

\frac{\frac{π}{4}}{4}

Apply the fraction rule:

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

= \frac{π}{4 \cdot 4}

Multiply the numbers:

4 \cdot 4 = 16

= \frac{π}{16}

= \frac{π}{4} + \frac{πn}{2} + \frac{π}{16} + \frac{arcsin ( - \frac{y + 3}{3} )}{4}

Group like terms

= \frac{π}{4} + \frac{π}{16} + \frac{πn}{2} + \frac{arcsin ( - \frac{y + 3}{3} )}{4}

x = \frac{π}{4} + \frac{π}{16} + \frac{πn}{2} + \frac{arcsin ( - \frac{y + 3}{3} )}{4}

x = \frac{π}{4} + \frac{π}{16} + \frac{πn}{2} + \frac{arcsin ( - \frac{y + 3}{3} )}{4}

x = \frac{π}{4} + \frac{π}{16} + \frac{πn}{2} + \frac{arcsin ( - \frac{y + 3}{3} )}{4}

x = \frac{arcsin ( \frac{y + 3}{3} )}{4} + \frac{πn}{2} + \frac{π}{16}, x = \frac{π}{4} + \frac{π}{16} + \frac{πn}{2} + \frac{arcsin ( - \frac{y + 3}{3} )}{4}

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