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solvefor x,h=4-2sin(3x+pi/3)

Solution

solve for

x, h = 4 - 2 sin (3 x + \frac{π}{3})

Solution

x = \frac{arcsin ( - \frac{h - 4}{2} )}{3} + \frac{2 πn}{3} - \frac{π}{9}, x = \frac{π}{3} - \frac{π}{9} + \frac{2 πn}{3} + \frac{arcsin ( \frac{h - 4}{2} )}{3}

Solution steps

h = 4 - 2 sin (3 x + \frac{π}{3})

Switch sides

4 - 2 sin (3 x + \frac{π}{3}) = h

Move

4

to the right side

4 - 2 sin (3 x + \frac{π}{3}) = h

Subtract

4

from both sides

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

Simplify

- 2 sin (3 x + \frac{π}{3}) = h - 4

- 2 sin (3 x + \frac{π}{3}) = h - 4

Divide both sides by

- 2

- 2 sin (3 x + \frac{π}{3}) = h - 4

Divide both sides by

- 2

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

Simplify

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

Simplify

\frac{- 2 sin ( 3 x + \frac{π}{3} )}{- 2} : sin (3 x + \frac{π}{3})

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

Apply the fraction rule:

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

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

Divide the numbers:

\frac{2}{2} = 1

= sin (3 x + \frac{π}{3})

Simplify

\frac{h}{- 2} - \frac{4}{- 2} : - \frac{h - 4}{2}

\frac{h}{- 2} - \frac{4}{- 2}

Apply rule

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

= \frac{h - 4}{- 2}

Apply the fraction rule:

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

= - \frac{h - 4}{2}

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

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

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

Apply trig inverse properties

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

General solutions for

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

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

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

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

Solve

3 x + \frac{π}{3} = arcsin (- \frac{h - 4}{2}) + 2 πn : x = \frac{arcsin ( - \frac{h - 4}{2} )}{3} + \frac{2 πn}{3} - \frac{π}{9}

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

Move

\frac{π}{3}

to the right side

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

Subtract

\frac{π}{3}

from both sides

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

Simplify

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

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

Divide both sides by

3

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

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{arcsin ( - \frac{h - 4}{2} )}{3} + \frac{2 πn}{3} - \frac{\frac{π}{3}}{3} : \frac{arcsin ( - \frac{h - 4}{2} )}{3} + \frac{2 πn}{3} - \frac{π}{9}

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

\frac{\frac{π}{3}}{3} = \frac{π}{9}

\frac{\frac{π}{3}}{3}

Apply the fraction rule:

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

= \frac{π}{3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{π}{9}

= \frac{arcsin ( - \frac{h - 4}{2} )}{3} + \frac{2 πn}{3} - \frac{π}{9}

x = \frac{arcsin ( - \frac{h - 4}{2} )}{3} + \frac{2 πn}{3} - \frac{π}{9}

x = \frac{arcsin ( - \frac{h - 4}{2} )}{3} + \frac{2 πn}{3} - \frac{π}{9}

x = \frac{arcsin ( - \frac{h - 4}{2} )}{3} + \frac{2 πn}{3} - \frac{π}{9}

Solve

3 x + \frac{π}{3} = π + arcsin (\frac{h - 4}{2}) + 2 πn : x = \frac{π}{3} - \frac{π}{9} + \frac{2 πn}{3} + \frac{arcsin ( \frac{h - 4}{2} )}{3}

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

Move

\frac{π}{3}

to the right side

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

Subtract

\frac{π}{3}

from both sides

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

Simplify

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

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

Divide both sides by

3

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

Divide both sides by

3

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

Simplify

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

Simplify

\frac{3 x}{3} : x

\frac{3 x}{3}

Divide the numbers:

\frac{3}{3} = 1

= x

Simplify

\frac{π}{3} + \frac{arcsin ( \frac{h - 4}{2} )}{3} + \frac{2 πn}{3} - \frac{\frac{π}{3}}{3} : \frac{π}{3} - \frac{π}{9} + \frac{2 πn}{3} + \frac{arcsin ( \frac{h - 4}{2} )}{3}

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

Group like terms

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

\frac{\frac{π}{3}}{3} = \frac{π}{9}

\frac{\frac{π}{3}}{3}

Apply the fraction rule:

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

= \frac{π}{3 \cdot 3}

Multiply the numbers:

3 \cdot 3 = 9

= \frac{π}{9}

= \frac{π}{3} + \frac{2 πn}{3} - \frac{π}{9} + \frac{arcsin ( \frac{h - 4}{2} )}{3}

Group like terms

= \frac{π}{3} - \frac{π}{9} + \frac{2 πn}{3} + \frac{arcsin ( \frac{h - 4}{2} )}{3}

x = \frac{π}{3} - \frac{π}{9} + \frac{2 πn}{3} + \frac{arcsin ( \frac{h - 4}{2} )}{3}

x = \frac{π}{3} - \frac{π}{9} + \frac{2 πn}{3} + \frac{arcsin ( \frac{h - 4}{2} )}{3}

x = \frac{π}{3} - \frac{π}{9} + \frac{2 πn}{3} + \frac{arcsin ( \frac{h - 4}{2} )}{3}

x = \frac{arcsin ( - \frac{h - 4}{2} )}{3} + \frac{2 πn}{3} - \frac{π}{9}, x = \frac{π}{3} - \frac{π}{9} + \frac{2 πn}{3} + \frac{arcsin ( \frac{h - 4}{2} )}{3}

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