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

Solution

solve for

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

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}

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

Apply trig inverse properties

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}

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}

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}

3 sin (θ) - 2 sin^{2} (θ) = 0

3 tan (\frac{x}{3}) + 3 = 0

0 = 2 sin (2 x)

sec^{2} (x) = tan (x) + 1

x, cos (x) - sin (2 x) = 0