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Popular >

solvefor y,z=e^{-x}sin(x+2y)

Solution

solve for

y, z = e^{- x} sin (x + 2 y)

Solution

y = \frac{arcsin ( e ^{x} z )}{2} + πn - \frac{x}{2}, y = \frac{π}{2} + \frac{arcsin ( - e ^{x} z )}{2} + πn - \frac{x}{2}

Solution steps

z = e^{- x} sin (x + 2 y)

Switch sides

e^{- x} sin (x + 2 y) = z

Divide both sides by

e^{- x}

e^{- x} sin (x + 2 y) = z

Divide both sides by

e^{- x}

\frac{e ^{- x} sin ( x + 2 y )}{e ^{- x}} = \frac{z}{e ^{- x}}

Simplify

\frac{e ^{- x} sin ( x + 2 y )}{e ^{- x}} = \frac{z}{e ^{- x}}

Simplify

\frac{e ^{- x} sin ( x + 2 y )}{e ^{- x}} : sin (x + 2 y)

\frac{e ^{- x} sin ( x + 2 y )}{e ^{- x}}

Cancel the common factor:

e^{- x}

= sin (x + 2 y)

Simplify

\frac{z}{e ^{- x}} : z e^{x}

\frac{z}{e ^{- x}}

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

e^{- x} = \frac{1}{e ^{x}}

= \frac{z}{\frac{1}{e ^{x}}}

Apply the fraction rule:

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

= \frac{z e ^{x}}{1}

Apply the fraction rule:

\frac{a}{1} = a

= z e^{x}

sin (x + 2 y) = z e^{x}

sin (x + 2 y) = z e^{x}

sin (x + 2 y) = z e^{x}

Apply trig inverse properties

sin (x + 2 y) = z e^{x}

General solutions for

sin (x + 2 y) = z e^{x}

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

x + 2 y = arcsin (z e^{x}) + 2 πn, x + 2 y = π + arcsin (- z e^{x}) + 2 πn

x + 2 y = arcsin (z e^{x}) + 2 πn, x + 2 y = π + arcsin (- z e^{x}) + 2 πn

Solve

x + 2 y = arcsin (z e^{x}) + 2 πn : y = \frac{arcsin ( e ^{x} z )}{2} + πn - \frac{x}{2}

x + 2 y = arcsin (z e^{x}) + 2 πn

Move

x

to the right side

x + 2 y = arcsin (z e^{x}) + 2 πn

Subtract

x

from both sides

x + 2 y - x = arcsin (z e^{x}) + 2 πn - x

Simplify

2 y = arcsin (z e^{x}) + 2 πn - x

2 y = arcsin (z e^{x}) + 2 πn - x

Divide both sides by

2

2 y = arcsin (z e^{x}) + 2 πn - x

Divide both sides by

2

\frac{2 y}{2} = \frac{arcsin ( z e ^{x} )}{2} + \frac{2 πn}{2} - \frac{x}{2}

Simplify

y = \frac{arcsin ( e ^{x} z )}{2} + πn - \frac{x}{2}

y = \frac{arcsin ( e ^{x} z )}{2} + πn - \frac{x}{2}

Solve

x + 2 y = π + arcsin (- z e^{x}) + 2 πn : y = \frac{π}{2} + \frac{arcsin ( - e ^{x} z )}{2} + πn - \frac{x}{2}

x + 2 y = π + arcsin (- z e^{x}) + 2 πn

Move

x

to the right side

x + 2 y = π + arcsin (- z e^{x}) + 2 πn

Subtract

x

from both sides

x + 2 y - x = π + arcsin (- z e^{x}) + 2 πn - x

Simplify

2 y = π + arcsin (- z e^{x}) + 2 πn - x

2 y = π + arcsin (- z e^{x}) + 2 πn - x

Divide both sides by

2

2 y = π + arcsin (- z e^{x}) + 2 πn - x

Divide both sides by

2

\frac{2 y}{2} = \frac{π}{2} + \frac{arcsin ( - z e ^{x} )}{2} + \frac{2 πn}{2} - \frac{x}{2}

Simplify

y = \frac{π}{2} + \frac{arcsin ( - e ^{x} z )}{2} + πn - \frac{x}{2}

y = \frac{π}{2} + \frac{arcsin ( - e ^{x} z )}{2} + πn - \frac{x}{2}

y = \frac{arcsin ( e ^{x} z )}{2} + πn - \frac{x}{2}, y = \frac{π}{2} + \frac{arcsin ( - e ^{x} z )}{2} + πn - \frac{x}{2}

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