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

The general solution for solvefor x,sin(y/x)+cos(x/y)=0 is

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

Solution

solve for

x, sin (\frac{y}{x}) + cos (\frac{x}{y}) = 0

Solution

x = - \frac{- π y - 4 πn y + ( π y + 4 πn y ) ^{2} + 16 y ^{2}}{4}, x = - \frac{- π y - 4 πn y - ( π y + 4 πn y ) ^{2} + 16 y ^{2}}{4}, x = \frac{- π y - 4 πn y + ( π y + 4 πn y ) ^{2} - 16 y ^{2}}{4}, x = \frac{- π y - 4 πn y - ( π y + 4 πn y ) ^{2} - 16 y ^{2}}{4}

Solution steps

sin (\frac{y}{x}) + cos (\frac{x}{y}) = 0

Subtract

sin (\frac{y}{x})

from both sides

cos (\frac{x}{y}) = - sin (\frac{y}{x})

Rewrite using trig identities

cos (\frac{x}{y}) = - sin (\frac{y}{x})

Use the following identity:

- sin (x) = sin (- x)

cos (\frac{x}{y}) = sin (- \frac{y}{x})

Use the following identity:

cos (x) = sin (\frac{π}{2} - x)

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

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

Apply trig inverse properties

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

sin (x) = sin (y) \Rightarrow x = y + 2 π n, x = π - y + 2 π n

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

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

- (\frac{y}{x}) = \frac{π}{2} - \frac{x}{y} + 2 πn : x = - \frac{- π y - 4 πn y + ( π y + 4 πn y ) ^{2} + 16 y ^{2}}{4}, x = - \frac{- π y - 4 πn y - ( π y + 4 πn y ) ^{2} + 16 y ^{2}}{4}

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

Multiply by LCM

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

Simplify

- (\frac{y}{x}) : - \frac{y}{x}

- (\frac{y}{x})

Remove parentheses:

(a) = a

= - \frac{y}{x}

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

Find Least Common Multiplier of

x, 2, y : 2 x y

x, 2, y

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear in at least one of the factored expressions

= 2 x y

Multiply by LCM=

2 x y

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

Simplify

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

Simplify

- \frac{y}{x} \cdot 2 x y : - 2 y^{2}

- \frac{y}{x} \cdot 2 x y

Multiply fractions:

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

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

Cancel the common factor:

x

= - y \cdot 2 y

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

yy = y^{1 + 1}

= - 2 y^{1 + 1}

Add the numbers:

1 + 1 = 2

= - 2 y^{2}

Simplify

\frac{π}{2} \cdot 2 x y : π x y

\frac{π}{2} \cdot 2 x y

Multiply fractions:

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

= \frac{2 π}{2} x y

Cancel the common factor:

2

= x y π

Simplify

- \frac{x}{y} \cdot 2 x y : - 2 x^{2}

- \frac{x}{y} \cdot 2 x y

Multiply fractions:

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

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

Cancel the common factor:

y

= - x \cdot 2 x

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

xx = x^{1 + 1}

= - 2 x^{1 + 1}

Add the numbers:

1 + 1 = 2

= - 2 x^{2}

Simplify

2 πn \cdot 2 x y : 4 πn x y

2 πn \cdot 2 x y

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn x y

- 2 y^{2} = π x y - 2 x^{2} + 4 πn x y

- 2 y^{2} = π x y - 2 x^{2} + 4 πn x y

- 2 y^{2} = π x y - 2 x^{2} + 4 πn x y

Solve

- 2 y^{2} = π x y - 2 x^{2} + 4 πn x y : x = - \frac{- π y - 4 πn y + ( π y + 4 πn y ) ^{2} + 16 y ^{2}}{4}, x = - \frac{- π y - 4 πn y - ( π y + 4 πn y ) ^{2} + 16 y ^{2}}{4}

- 2 y^{2} = π x y - 2 x^{2} + 4 πn x y

Switch sides

π x y - 2 x^{2} + 4 πn x y = - 2 y^{2}

Move

2 y^{2}

to the left side

π x y - 2 x^{2} + 4 πn x y = - 2 y^{2}

Add

2 y^{2}

to both sides

π x y - 2 x^{2} + 4 πn x y + 2 y^{2} = - 2 y^{2} + 2 y^{2}

Simplify

π x y - 2 x^{2} + 4 πn x y + 2 y^{2} = 0

π x y - 2 x^{2} + 4 πn x y + 2 y^{2} = 0

Write in the standard form

a x^{2} + b x + c = 0

- 2 x^{2} + (π y + 4 πn y) x + 2 y^{2} = 0

Solve with the quadratic formula

- 2 x^{2} + (π y + 4 πn y) x + 2 y^{2} = 0

Quadratic Equation Formula:

For

a = - 2, b = π y + 4 πn y, c = 2 y^{2}

x_{1, 2} = \frac{- ( π y + 4 πn y ) \pm ( π y + 4 πn y ) ^{2} - 4 ( - 2 ) \cdot 2 y ^{2}}{2 ( - 2 )}

x_{1, 2} = \frac{- ( π y + 4 πn y ) \pm ( π y + 4 πn y ) ^{2} - 4 ( - 2 ) \cdot 2 y ^{2}}{2 ( - 2 )}

Simplify

(π y + 4 πn y)^{2} - 4 (- 2) \cdot 2 y^{2} : (π y + 4 πn y)^{2} + 16 y^{2}

(π y + 4 πn y)^{2} - 4 (- 2) \cdot 2 y^{2}

Apply rule

- (- a) = a

= (π y + 4 πn y)^{2} + 4 \cdot 2 \cdot 2 y^{2}

Multiply the numbers:

4 \cdot 2 \cdot 2 = 16

= (π y + 4 πn y)^{2} + 16 y^{2}

x_{1, 2} = \frac{- ( π y + 4 πn y ) \pm ( π y + 4 πn y ) ^{2} + 16 y ^{2}}{2 ( - 2 )}

Separate the solutions

x_{1} = \frac{- ( π y + 4 πn y ) + ( π y + 4 πn y ) ^{2} + 16 y ^{2}}{2 ( - 2 )}, x_{2} = \frac{- ( π y + 4 πn y ) - ( π y + 4 πn y ) ^{2} + 16 y ^{2}}{2 ( - 2 )}

x = \frac{- ( π y + 4 πn y ) + ( π y + 4 πn y ) ^{2} + 16 y ^{2}}{2 ( - 2 )} : - \frac{- π y - 4 πn y + ( π y + 4 πn y ) ^{2} + 16 y ^{2}}{4}

\frac{- ( π y + 4 πn y ) + ( π y + 4 πn y ) ^{2} + 16 y ^{2}}{2 ( - 2 )}

Remove parentheses:

(- a) = - a

= \frac{- ( π y + 4 πn y ) + ( π y + 4 πn y ) ^{2} + 16 y ^{2}}{- 2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- ( π y + 4 πn y ) + 16 y ^{2} + ( π y + 4 πn y ) ^{2}}{- 4}

Apply the fraction rule:

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

= - \frac{- ( π y + 4 πn y ) + ( π y + 4 πn y ) ^{2} + 16 y ^{2}}{4}

- (π y + 4 πn y) : - π y - 4 πn y

- (π y + 4 πn y)

Distribute parentheses

= - π y - 4 πn y

Apply minus-plus rules

+ (- a) = - a

= - π y - 4 πn y

= - \frac{- π y - 4 πn y + 16 y ^{2} + ( π y + 4 πn y ) ^{2}}{4}

= - \frac{- π y - 4 πn y + ( π y + 4 πn y ) ^{2} + 16 y ^{2}}{4}

x = \frac{- ( π y + 4 πn y ) - ( π y + 4 πn y ) ^{2} + 16 y ^{2}}{2 ( - 2 )} : - \frac{- π y - 4 πn y - ( π y + 4 πn y ) ^{2} + 16 y ^{2}}{4}

\frac{- ( π y + 4 πn y ) - ( π y + 4 πn y ) ^{2} + 16 y ^{2}}{2 ( - 2 )}

Remove parentheses:

(- a) = - a

= \frac{- ( π y + 4 πn y ) - ( π y + 4 πn y ) ^{2} + 16 y ^{2}}{- 2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- ( π y + 4 πn y ) - 16 y ^{2} + ( π y + 4 πn y ) ^{2}}{- 4}

Apply the fraction rule:

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

= - \frac{- ( π y + 4 πn y ) - ( π y + 4 πn y ) ^{2} + 16 y ^{2}}{4}

- (π y + 4 πn y) : - π y - 4 πn y

- (π y + 4 πn y)

Distribute parentheses

= - π y - 4 πn y

Apply minus-plus rules

+ (- a) = - a

= - π y - 4 πn y

= - \frac{- π y - 4 πn y - 16 y ^{2} + ( π y + 4 πn y ) ^{2}}{4}

= - \frac{- π y - 4 πn y - ( π y + 4 πn y ) ^{2} + 16 y ^{2}}{4}

The solutions to the quadratic equation are:

x = - \frac{- π y - 4 πn y + ( π y + 4 πn y ) ^{2} + 16 y ^{2}}{4}, x = - \frac{- π y - 4 πn y - ( π y + 4 πn y ) ^{2} + 16 y ^{2}}{4}

x = - \frac{- π y - 4 πn y + ( π y + 4 πn y ) ^{2} + 16 y ^{2}}{4}, x = - \frac{- π y - 4 πn y - ( π y + 4 πn y ) ^{2} + 16 y ^{2}}{4}

- (\frac{y}{x}) = π - (\frac{π}{2} - \frac{x}{y}) + 2 πn : x = \frac{- π y - 4 πn y + ( π y + 4 πn y ) ^{2} - 16 y ^{2}}{4}, x = \frac{- π y - 4 πn y - ( π y + 4 πn y ) ^{2} - 16 y ^{2}}{4}

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

Multiply by LCM

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

Simplify

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

Simplify

- (\frac{y}{x}) : - \frac{y}{x}

- (\frac{y}{x})

Remove parentheses:

(a) = a

= - \frac{y}{x}

Simplify

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

- (\frac{π}{2} - \frac{x}{y})

Distribute parentheses

= - \frac{π}{2} - (- \frac{x}{y})

Apply minus-plus rules

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

= - \frac{π}{2} + \frac{x}{y}

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

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

Find Least Common Multiplier of

x, 2, y : 2 x y

x, 2, y

Lowest Common Multiplier (LCM)

Compute an expression comprised of factors that appear in at least one of the factored expressions

= 2 x y

Multiply by LCM=

2 x y

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

Simplify

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

Simplify

- \frac{y}{x} \cdot 2 x y : - 2 y^{2}

- \frac{y}{x} \cdot 2 x y

Multiply fractions:

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

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

Cancel the common factor:

x

= - y \cdot 2 y

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

yy = y^{1 + 1}

= - 2 y^{1 + 1}

Add the numbers:

1 + 1 = 2

= - 2 y^{2}

Simplify

- \frac{π}{2} \cdot 2 x y : - π x y

- \frac{π}{2} \cdot 2 x y

Multiply fractions:

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

= - \frac{2 π}{2} x y

Cancel the common factor:

2

= - x y π

Simplify

\frac{x}{y} \cdot 2 x y : 2 x^{2}

\frac{x}{y} \cdot 2 x y

Multiply fractions:

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

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

Cancel the common factor:

y

= x \cdot 2 x

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

xx = x^{1 + 1}

= 2 x^{1 + 1}

Add the numbers:

1 + 1 = 2

= 2 x^{2}

Simplify

2 πn \cdot 2 x y : 4 πn x y

2 πn \cdot 2 x y

Multiply the numbers:

2 \cdot 2 = 4

= 4 πn x y

- 2 y^{2} = π 2 x y - π x y + 2 x^{2} + 4 πn x y

Simplify

π 2 x y - π x y + 2 x^{2} + 4 πn x y : π x y + 2 x^{2} + 4 πn x y

π 2 x y - π x y + 2 x^{2} + 4 πn x y

Add similar elements:

2 π x y - π x y = π x y

= π x y + 2 x^{2} + 4 πn x y

- 2 y^{2} = π x y + 2 x^{2} + 4 πn x y

- 2 y^{2} = π x y + 2 x^{2} + 4 πn x y

- 2 y^{2} = π x y + 2 x^{2} + 4 πn x y

Solve

- 2 y^{2} = π x y + 2 x^{2} + 4 πn x y : x = \frac{- π y - 4 πn y + ( π y + 4 πn y ) ^{2} - 16 y ^{2}}{4}, x = \frac{- π y - 4 πn y - ( π y + 4 πn y ) ^{2} - 16 y ^{2}}{4}

- 2 y^{2} = π x y + 2 x^{2} + 4 πn x y

Switch sides

π x y + 2 x^{2} + 4 πn x y = - 2 y^{2}

Move

2 y^{2}

to the left side

π x y + 2 x^{2} + 4 πn x y = - 2 y^{2}

Add

2 y^{2}

to both sides

π x y + 2 x^{2} + 4 πn x y + 2 y^{2} = - 2 y^{2} + 2 y^{2}

Simplify

π x y + 2 x^{2} + 4 πn x y + 2 y^{2} = 0

π x y + 2 x^{2} + 4 πn x y + 2 y^{2} = 0

Write in the standard form

a x^{2} + b x + c = 0

2 x^{2} + (π y + 4 πn y) x + 2 y^{2} = 0

Solve with the quadratic formula

2 x^{2} + (π y + 4 πn y) x + 2 y^{2} = 0

Quadratic Equation Formula:

For

a = 2, b = π y + 4 πn y, c = 2 y^{2}

x_{1, 2} = \frac{- ( π y + 4 πn y ) \pm ( π y + 4 πn y ) ^{2} - 4 \cdot 2 \cdot 2 y ^{2}}{2 \cdot 2}

x_{1, 2} = \frac{- ( π y + 4 πn y ) \pm ( π y + 4 πn y ) ^{2} - 4 \cdot 2 \cdot 2 y ^{2}}{2 \cdot 2}

Simplify

(π y + 4 πn y)^{2} - 4 \cdot 2 \cdot 2 y^{2} : (π y + 4 πn y)^{2} - 16 y^{2}

(π y + 4 πn y)^{2} - 4 \cdot 2 \cdot 2 y^{2}

Multiply the numbers:

4 \cdot 2 \cdot 2 = 16

= (π y + 4 πn y)^{2} - 16 y^{2}

x_{1, 2} = \frac{- ( π y + 4 πn y ) \pm ( π y + 4 πn y ) ^{2} - 16 y ^{2}}{2 \cdot 2}

Separate the solutions

x_{1} = \frac{- ( π y + 4 πn y ) + ( π y + 4 πn y ) ^{2} - 16 y ^{2}}{2 \cdot 2}, x_{2} = \frac{- ( π y + 4 πn y ) - ( π y + 4 πn y ) ^{2} - 16 y ^{2}}{2 \cdot 2}

x = \frac{- ( π y + 4 πn y ) + ( π y + 4 πn y ) ^{2} - 16 y ^{2}}{2 \cdot 2} : \frac{- π y - 4 πn y + ( π y + 4 πn y ) ^{2} - 16 y ^{2}}{4}

\frac{- ( π y + 4 πn y ) + ( π y + 4 πn y ) ^{2} - 16 y ^{2}}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- ( π y + 4 πn y ) + - 16 y ^{2} + ( π y + 4 πn y ) ^{2}}{4}

- (π y + 4 πn y) : - π y - 4 πn y

- (π y + 4 πn y)

Distribute parentheses

= - π y - 4 πn y

Apply minus-plus rules

+ (- a) = - a

= - π y - 4 πn y

= \frac{- π y - 4 πn y + - 16 y ^{2} + ( π y + 4 πn y ) ^{2}}{4}

= \frac{- π y - 4 πn y + ( π y + 4 πn y ) ^{2} - 16 y ^{2}}{4}

x = \frac{- ( π y + 4 πn y ) - ( π y + 4 πn y ) ^{2} - 16 y ^{2}}{2 \cdot 2} : \frac{- π y - 4 πn y - ( π y + 4 πn y ) ^{2} - 16 y ^{2}}{4}

\frac{- ( π y + 4 πn y ) - ( π y + 4 πn y ) ^{2} - 16 y ^{2}}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{- ( π y + 4 πn y ) - - 16 y ^{2} + ( π y + 4 πn y ) ^{2}}{4}

- (π y + 4 πn y) : - π y - 4 πn y

- (π y + 4 πn y)

Distribute parentheses

= - π y - 4 πn y

Apply minus-plus rules

+ (- a) = - a

= - π y - 4 πn y

= \frac{- π y - 4 πn y - - 16 y ^{2} + ( π y + 4 πn y ) ^{2}}{4}

= \frac{- π y - 4 πn y - ( π y + 4 πn y ) ^{2} - 16 y ^{2}}{4}

The solutions to the quadratic equation are:

x = \frac{- π y - 4 πn y + ( π y + 4 πn y ) ^{2} - 16 y ^{2}}{4}, x = \frac{- π y - 4 πn y - ( π y + 4 πn y ) ^{2} - 16 y ^{2}}{4}

x = \frac{- π y - 4 πn y + ( π y + 4 πn y ) ^{2} - 16 y ^{2}}{4}, x = \frac{- π y - 4 πn y - ( π y + 4 πn y ) ^{2} - 16 y ^{2}}{4}

x = - \frac{- π y - 4 πn y + ( π y + 4 πn y ) ^{2} + 16 y ^{2}}{4}, x = - \frac{- π y - 4 πn y - ( π y + 4 πn y ) ^{2} + 16 y ^{2}}{4}, x = \frac{- π y - 4 πn y + ( π y + 4 πn y ) ^{2} - 16 y ^{2}}{4}, x = \frac{- π y - 4 πn y - ( π y + 4 πn y ) ^{2} - 16 y ^{2}}{4}

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Frequently Asked Questions (FAQ)

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