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

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

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

Solution

solve for

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

Solution

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

Solution steps

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

Rewrite using trig identities

cos (y + x) + sin (x)

Use the following identity:

sin (x) = cos (\frac{π}{2} - x)

= cos (y + x) + cos (\frac{π}{2} - x)

Use the Sum to Product identity:

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

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

Simplify

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

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

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

\frac{\frac{π}{2} - x + y + x}{2}

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

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

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

\frac{\frac{π}{2} - x - ( y + x )}{2}

Join

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

\frac{π}{2} - x - (y + x)

Convert element to fraction:

x = \frac{x 2}{2}, (x + y) = \frac{( y + x ) 2}{2}

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

Expand

π - x \cdot 2 - (y + x) \cdot 2 : - 4 x + π - 2 y

π - x \cdot 2 - (y + x) \cdot 2

= π - 2 x - 2 (y + x)

Expand

- 2 (y + x) : - 2 y - 2 x

- 2 (y + x)

Apply the distributive law:

a (b + c) = ab + a c

a = - 2, b = y, c = x

= - 2 y + (- 2) x

Apply minus-plus rules

+ (- a) = - a

= - 2 y - 2 x

= π - x \cdot 2 - 2 y - 2 x

Simplify

π - x \cdot 2 - 2 y - 2 x : - 4 x + π - 2 y

π - x \cdot 2 - 2 y - 2 x

Group like terms

= - 2 x - 2 x + π - 2 y

Add similar elements:

- 2 x - 2 x = - 4 x

= - 4 x + π - 2 y

= - 4 x + π - 2 y

= \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 - 2 y}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- 4 x + π - 2 y}{4}

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

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

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

Using the Zero Factor Principle:

ab = 0

then

a = 0

b = 0

cos (\frac{- 4 x + π - 2 y}{4}) = 0

General solutions for

cos (\frac{- 4 x + π - 2 y}{4}) = 0

cos (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} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{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} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

\frac{- 4 x + π - 2 y}{4} = \frac{π}{2} + 2 πn, \frac{- 4 x + π - 2 y}{4} = \frac{3 π}{2} + 2 πn

\frac{- 4 x + π - 2 y}{4} = \frac{π}{2} + 2 πn, \frac{- 4 x + π - 2 y}{4} = \frac{3 π}{2} + 2 πn

Solve

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

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

Multiply both sides by

4

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

Multiply both sides by

4

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

Simplify

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

Simplify

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

\frac{4 ( - 4 x + π - 2 y )}{4}

Divide the numbers:

\frac{4}{4} = 1

= - 4 x + π - 2 y

Simplify

4 \cdot \frac{π}{2} + 4 \cdot 2 πn : 2 π + 8 πn

4 \cdot \frac{π}{2} + 4 \cdot 2 πn

4 \cdot \frac{π}{2} = 2 π

4 \cdot \frac{π}{2}

Multiply fractions:

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

= \frac{π 4}{2}

Divide the numbers:

\frac{4}{2} = 2

= 2 π

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= 8 πn

= 2 π + 8 πn

- 4 x + π - 2 y = 2 π + 8 πn

- 4 x + π - 2 y = 2 π + 8 πn

- 4 x + π - 2 y = 2 π + 8 πn

Move

2 y

to the right side

- 4 x + π - 2 y = 2 π + 8 πn

Add

2 y

to both sides

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

Simplify

- 4 x + π = 2 π + 8 πn + 2 y

- 4 x + π = 2 π + 8 πn + 2 y

Move

π

to the right side

- 4 x + π = 2 π + 8 πn + 2 y

Subtract

π

from both sides

- 4 x + π - π = 2 π + 8 πn + 2 y - π

Simplify

- 4 x = 2 y + π + 8 πn

- 4 x = 2 y + π + 8 πn

Divide both sides by

- 4

- 4 x = 2 y + π + 8 πn

Divide both sides by

- 4

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

Simplify

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

Simplify

\frac{- 4 x}{- 4} : x

\frac{- 4 x}{- 4}

Apply the fraction rule:

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

= \frac{4 x}{4}

Divide the numbers:

\frac{4}{4} = 1

= x

Simplify

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

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

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

\frac{2 y}{- 4}

Apply the fraction rule:

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

= - \frac{2 y}{4}

Cancel the common factor:

2

= - \frac{y}{2}

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

Apply the fraction rule:

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

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

\frac{8 πn}{- 4} = - 2 πn

\frac{8 πn}{- 4}

Apply the fraction rule:

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

= - \frac{8 πn}{4}

Divide the numbers:

\frac{8}{4} = 2

= - 2 πn

= - \frac{y}{2} - \frac{π}{4} - 2 πn

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

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

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

Solve

\frac{- 4 x + π - 2 y}{4} = \frac{3 π}{2} + 2 πn : x = - \frac{y}{2} - \frac{5 π}{4} - 2 πn

\frac{- 4 x + π - 2 y}{4} = \frac{3 π}{2} + 2 πn

Multiply both sides by

4

\frac{- 4 x + π - 2 y}{4} = \frac{3 π}{2} + 2 πn

Multiply both sides by

4

\frac{4 ( - 4 x + π - 2 y )}{4} = 4 \cdot \frac{3 π}{2} + 4 \cdot 2 πn

Simplify

\frac{4 ( - 4 x + π - 2 y )}{4} = 4 \cdot \frac{3 π}{2} + 4 \cdot 2 πn

Simplify

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

\frac{4 ( - 4 x + π - 2 y )}{4}

Divide the numbers:

\frac{4}{4} = 1

= - 4 x + π - 2 y

Simplify

4 \cdot \frac{3 π}{2} + 4 \cdot 2 πn : 6 π + 8 πn

4 \cdot \frac{3 π}{2} + 4 \cdot 2 πn

4 \cdot \frac{3 π}{2} = 6 π

4 \cdot \frac{3 π}{2}

Multiply fractions:

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

= \frac{3 π 4}{2}

Multiply the numbers:

3 \cdot 4 = 12

= \frac{12 π}{2}

Divide the numbers:

\frac{12}{2} = 6

= 6 π

4 \cdot 2 πn = 8 πn

4 \cdot 2 πn

Multiply the numbers:

4 \cdot 2 = 8

= 8 πn

= 6 π + 8 πn

- 4 x + π - 2 y = 6 π + 8 πn

- 4 x + π - 2 y = 6 π + 8 πn

- 4 x + π - 2 y = 6 π + 8 πn

Move

2 y

to the right side

- 4 x + π - 2 y = 6 π + 8 πn

Add

2 y

to both sides

- 4 x + π - 2 y + 2 y = 6 π + 8 πn + 2 y

Simplify

- 4 x + π = 6 π + 8 πn + 2 y

- 4 x + π = 6 π + 8 πn + 2 y

Move

π

to the right side

- 4 x + π = 6 π + 8 πn + 2 y

Subtract

π

from both sides

- 4 x + π - π = 6 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}

Apply the fraction rule:

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

= \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}

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

\frac{2 y}{- 4}

Apply the fraction rule:

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

= - \frac{2 y}{4}

Cancel the common factor:

2

= - \frac{y}{2}

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

Apply the fraction rule:

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

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

\frac{8 πn}{- 4} = - 2 πn

\frac{8 πn}{- 4}

Apply the fraction rule:

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

= - \frac{8 πn}{4}

Divide the numbers:

\frac{8}{4} = 2

= - 2 πn

= - \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 = - \frac{y}{2} - \frac{π}{4} - 2 πn, 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+y)+sin(x)=0 ?
The general solution for solvefor x,cos(x+y)+sin(x)=0 is x=-y/2-pi/4-2pin,x=-y/2-(5pi)/4-2pin