What is the general solution for solvefor x,cos(x)-sin(x+y)=0 ?

The general solution for solvefor x,cos(x)-sin(x+y)=0 is x=2pin-y/2+pi/4 ,x=-y/2+(5pi)/4+2pin

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solvefor x,cos(x)-sin(x+y)=0

Solution

solve for

x, cos (x) - sin (x + y) = 0

Solution

x = 2 πn - \frac{y}{2} + \frac{π}{4}, x = - \frac{y}{2} + \frac{5 π}{4} + 2 πn

Solution steps

cos (x) - sin (x + y) = 0

Rewrite using trig identities

cos (x) - sin (y + x)

Use the following identity:

sin (x) = cos (\frac{π}{2} - x)

= cos (x) - cos (\frac{π}{2} - (y + x))

- (y + x) : - y - x

- (y + x)

Distribute parentheses

= - y - x

Apply minus-plus rules

+ (- a) = - a

= - y - x

= cos (x) - cos (\frac{π}{2} - y - x)

Use the Sum to Product identity:

cos (s) - cos (t) = - 2 sin (\frac{s + t}{2}) sin (\frac{s - t}{2})

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

Simplify

- 2 sin (\frac{x + \frac{π}{2} - y - x}{2}) sin (\frac{x - ( \frac{π}{2} - y - x )}{2}) : - 2 sin (\frac{π - 2 y}{4}) sin (\frac{4 x - π + 2 y}{4})

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

\frac{x + \frac{π}{2} - y - x}{2} = \frac{π - 2 y}{4}

\frac{x + \frac{π}{2} - y - x}{2}

x + \frac{π}{2} - y - x = \frac{π}{2} - y

x + \frac{π}{2} - y - x

Group like terms

= x - x + \frac{π}{2} - y

Add similar elements:

x - x = 0

= \frac{π}{2} - y

= \frac{\frac{π}{2} - y}{2}

Join

\frac{π}{2} - y : \frac{π - 2 y}{2}

\frac{π}{2} - y

Convert element to fraction:

y = \frac{y 2}{2}

= \frac{π}{2} - \frac{y \cdot 2}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{π - y \cdot 2}{2}

= \frac{\frac{π - 2 y}{2}}{2}

Apply the fraction rule:

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

= \frac{π - y \cdot 2}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π - 2 y}{4}

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

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

\frac{x - ( \frac{π}{2} - y - x )}{2}

Expand

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

x - (\frac{π}{2} - y - x)

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

- (\frac{π}{2} - y - x)

Distribute parentheses

= - \frac{π}{2} - (- y) - (- x)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - \frac{π}{2} + y + x

= x - \frac{π}{2} + y + x

Simplify

x - \frac{π}{2} + y + x : 2 x - \frac{π}{2} + y

x - \frac{π}{2} + y + x

Group like terms

= x + x - \frac{π}{2} + y

Add similar elements:

x + x = 2 x

= 2 x - \frac{π}{2} + y

= 2 x - \frac{π}{2} + y

= \frac{2 x - \frac{π}{2} + y}{2}

Join

2 x - \frac{π}{2} + y : \frac{4 x - π + 2 y}{2}

2 x - \frac{π}{2} + y

Convert element to fraction:

2 x = \frac{2 x 2}{2}, y = \frac{y 2}{2}

= \frac{2 x \cdot 2}{2} - \frac{π}{2} + \frac{y \cdot 2}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{2 x \cdot 2 - π + y \cdot 2}{2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 x - π + 2 y}{2}

= \frac{\frac{4 x - π + 2 y}{2}}{2}

Apply the fraction rule:

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

= \frac{4 x - π + y \cdot 2}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 x - π + 2 y}{4}

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

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

- 2 sin (\frac{π - 2 y}{4}) sin (\frac{4 x - π + 2 y}{4}) = 0

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

sin (\frac{4 x - π + 2 y}{4}) = 0

General solutions for

sin (\frac{4 x - π + 2 y}{4}) = 0

sin (x)

periodicity table with

2 πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

\frac{4 x - π + 2 y}{4} = 0 + 2 πn, \frac{4 x - π + 2 y}{4} = π + 2 πn

\frac{4 x - π + 2 y}{4} = 0 + 2 πn, \frac{4 x - π + 2 y}{4} = π + 2 πn

Solve

\frac{4 x - π + 2 y}{4} = 0 + 2 πn : x = 2 πn - \frac{y}{2} + \frac{π}{4}

\frac{4 x - π + 2 y}{4} = 0 + 2 πn

0 + 2 πn = 2 πn

\frac{4 x - π + 2 y}{4} = 2 πn

Multiply both sides by

4

\frac{4 x - π + 2 y}{4} = 2 πn

Multiply both sides by

4

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

Simplify

4 x - π + 2 y = 8 πn

4 x - π + 2 y = 8 πn

Move

2 y

to the right side

4 x - π + 2 y = 8 πn

Subtract

2 y

from both sides

4 x - π + 2 y - 2 y = 8 πn - 2 y

Simplify

4 x - π = 8 πn - 2 y

4 x - π = 8 πn - 2 y

Move

π

to the right side

4 x - π = 8 πn - 2 y

Add

π

to both sides

4 x - π + π = 8 πn - 2 y + π

Simplify

4 x = 8 πn - 2 y + π

4 x = 8 πn - 2 y + π

Divide both sides by

4

4 x = 8 πn - 2 y + π

Divide both sides by

4

\frac{4 x}{4} = \frac{8 πn}{4} - \frac{2 y}{4} + \frac{π}{4}

Simplify

\frac{4 x}{4} = \frac{8 πn}{4} - \frac{2 y}{4} + \frac{π}{4}

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

\frac{8 πn}{4} - \frac{2 y}{4} + \frac{π}{4} : 2 πn - \frac{y}{2} + \frac{π}{4}

\frac{8 πn}{4} - \frac{2 y}{4} + \frac{π}{4}

Divide the numbers:

\frac{8}{4} = 2

= 2 πn - \frac{2 y}{4} + \frac{π}{4}

Cancel

\frac{2 y}{4} : \frac{y}{2}

\frac{2 y}{4}

Cancel the common factor:

2

= \frac{y}{2}

= 2 πn - \frac{y}{2} + \frac{π}{4}

x = 2 πn - \frac{y}{2} + \frac{π}{4}

x = 2 πn - \frac{y}{2} + \frac{π}{4}

x = 2 πn - \frac{y}{2} + \frac{π}{4}

Solve

\frac{4 x - π + 2 y}{4} = π + 2 πn : x = - \frac{y}{2} + \frac{5 π}{4} + 2 πn

\frac{4 x - π + 2 y}{4} = π + 2 πn

Multiply both sides by

4

\frac{4 x - π + 2 y}{4} = π + 2 πn

Multiply both sides by

4

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

Simplify

4 x - π + 2 y = 4 π + 8 πn

4 x - π + 2 y = 4 π + 8 πn

Move

2 y

to the right side

4 x - π + 2 y = 4 π + 8 πn

Subtract

2 y

from both sides

4 x - π + 2 y - 2 y = 4 π + 8 πn - 2 y

Simplify

4 x - π = 4 π + 8 πn - 2 y

4 x - π = 4 π + 8 πn - 2 y

Move

π

to the right side

4 x - π = 4 π + 8 πn - 2 y

Add

π

to both sides

4 x - π + π = 4 π + 8 πn - 2 y + π

Simplify

4 x = - 2 y + 5 π + 8 πn

4 x = - 2 y + 5 π + 8 πn

Divide both sides by

4

4 x = - 2 y + 5 π + 8 πn

Divide both sides by

4

\frac{4 x}{4} = - \frac{2 y}{4} + \frac{5 π}{4} + \frac{8 πn}{4}

Simplify

\frac{4 x}{4} = - \frac{2 y}{4} + \frac{5 π}{4} + \frac{8 πn}{4}

Simplify

\frac{4 x}{4} : x

\frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

- \frac{2 y}{4} + \frac{5 π}{4} + \frac{8 πn}{4} : - \frac{y}{2} + \frac{5 π}{4} + 2 πn

- \frac{2 y}{4} + \frac{5 π}{4} + \frac{8 πn}{4}

Cancel

\frac{2 y}{4} : \frac{y}{2}

\frac{2 y}{4}

Cancel the common factor:

2

= \frac{y}{2}

= - \frac{y}{2} + \frac{5 π}{4} + \frac{8 πn}{4}

Divide the numbers:

\frac{8}{4} = 2

= - \frac{y}{2} + \frac{5 π}{4} + 2 πn

x = - \frac{y}{2} + \frac{5 π}{4} + 2 πn

x = - \frac{y}{2} + \frac{5 π}{4} + 2 πn

x = - \frac{y}{2} + \frac{5 π}{4} + 2 πn

x = 2 πn - \frac{y}{2} + \frac{π}{4}, x = - \frac{y}{2} + \frac{5 π}{4} + 2 πn

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Frequently Asked Questions (FAQ)

What is the general solution for solvefor x,cos(x)-sin(x+y)=0 ?
The general solution for solvefor x,cos(x)-sin(x+y)=0 is x=2pin-y/2+pi/4 ,x=-y/2+(5pi)/4+2pin