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Popular >

solvefor x,e^ycos(2x)+y=1

Solution

solve for

x, e^{y} cos (2 x) + y = 1

Solution

x = \frac{arccos ( \frac{1 - y}{e ^{y}} )}{2} + πn, x = - \frac{arccos ( \frac{1 - y}{e ^{y}} )}{2} + πn

Solution steps

e^{y} cos (2 x) + y = 1

Move

y

to the right side

e^{y} cos (2 x) + y = 1

Subtract

y

from both sides

e^{y} cos (2 x) + y - y = 1 - y

Simplify

e^{y} cos (2 x) = 1 - y

e^{y} cos (2 x) = 1 - y

Divide both sides by

e^{y}

e^{y} cos (2 x) = 1 - y

Divide both sides by

e^{y}

\frac{e ^{y} cos ( 2 x )}{e ^{y}} = \frac{1}{e ^{y}} - \frac{y}{e ^{y}}

Simplify

\frac{e ^{y} cos ( 2 x )}{e ^{y}} = \frac{1}{e ^{y}} - \frac{y}{e ^{y}}

Simplify

\frac{e ^{y} cos ( 2 x )}{e ^{y}} : cos (2 x)

\frac{e ^{y} cos ( 2 x )}{e ^{y}}

Cancel the common factor:

e^{y}

= cos (2 x)

Simplify

\frac{1}{e ^{y}} - \frac{y}{e ^{y}} : \frac{1 - y}{e ^{y}}

\frac{1}{e ^{y}} - \frac{y}{e ^{y}}

Apply rule

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

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

cos (2 x) = \frac{1 - y}{e ^{y}}

cos (2 x) = \frac{1 - y}{e ^{y}}

cos (2 x) = \frac{1 - y}{e ^{y}}

Apply trig inverse properties

cos (2 x) = \frac{1 - y}{e ^{y}}

General solutions for

cos (2 x) = \frac{1 - y}{e ^{y}}

cos (x) = a \Rightarrow x = arccos (a) + 2 πn, x = - arccos (a) + 2 πn

2 x = arccos (\frac{1 - y}{e ^{y}}) + 2 πn, 2 x = - arccos (\frac{1 - y}{e ^{y}}) + 2 πn

2 x = arccos (\frac{1 - y}{e ^{y}}) + 2 πn, 2 x = - arccos (\frac{1 - y}{e ^{y}}) + 2 πn

Solve

2 x = arccos (\frac{1 - y}{e ^{y}}) + 2 πn : x = \frac{arccos ( \frac{1 - y}{e ^{y}} )}{2} + πn

2 x = arccos (\frac{1 - y}{e ^{y}}) + 2 πn

Divide both sides by

2

2 x = arccos (\frac{1 - y}{e ^{y}}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = \frac{arccos ( \frac{1 - y}{e ^{y}} )}{2} + \frac{2 πn}{2}

Simplify

x = \frac{arccos ( \frac{1 - y}{e ^{y}} )}{2} + πn

x = \frac{arccos ( \frac{1 - y}{e ^{y}} )}{2} + πn

Solve

2 x = - arccos (\frac{1 - y}{e ^{y}}) + 2 πn : x = - \frac{arccos ( \frac{1 - y}{e ^{y}} )}{2} + πn

2 x = - arccos (\frac{1 - y}{e ^{y}}) + 2 πn

Divide both sides by

2

2 x = - arccos (\frac{1 - y}{e ^{y}}) + 2 πn

Divide both sides by

2

\frac{2 x}{2} = - \frac{arccos ( \frac{1 - y}{e ^{y}} )}{2} + \frac{2 πn}{2}

Simplify

x = - \frac{arccos ( \frac{1 - y}{e ^{y}} )}{2} + πn

x = - \frac{arccos ( \frac{1 - y}{e ^{y}} )}{2} + πn

x = \frac{arccos ( \frac{1 - y}{e ^{y}} )}{2} + πn, x = - \frac{arccos ( \frac{1 - y}{e ^{y}} )}{2} + πn

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