What is the general solution for solvefor x,cos(xy)-sin(xy)=2 ?

The general solution for solvefor x,cos(xy)-sin(xy)=2 is No Solution for x\in\mathbb{R}

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solvefor x,cos(xy)-sin(xy)=2

Solution

solve for

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

Solution

No Solution for x \in R

Solution steps

cos (x y) - sin (x y) = 2

Subtract

2

from both sides

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

Rewrite using trig identities

- 2 + cos (y x) - sin (y x)

Use the following identity:

sin (x) = cos (\frac{π}{2} - x)

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

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

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

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

\frac{y x + \frac{π}{2} - y x}{2}

y x + \frac{π}{2} - y x = \frac{π}{2}

y x + \frac{π}{2} - y x

Group like terms

= x y - x y + \frac{π}{2}

Add similar elements:

x y - x y = 0

= \frac{π}{2}

= \frac{\frac{π}{2}}{2}

Apply the fraction rule:

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

= \frac{π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{π}{4}

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

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

\frac{y x - ( \frac{π}{2} - y x )}{2}

Expand

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

y x - (\frac{π}{2} - y x)

= x y - (\frac{π}{2} - x y)

- (\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

= y x - \frac{π}{2} + y x

Simplify

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

y x - \frac{π}{2} + y x

Group like terms

= x y + x y - \frac{π}{2}

Add similar elements:

x y + x y = 2 x y

= 2 x y - \frac{π}{2}

= 2 x y - \frac{π}{2}

= \frac{2 x y - \frac{π}{2}}{2}

Join

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

2 x y - \frac{π}{2}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

= \frac{2 x y \cdot 2 - π}{2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 x y - π}{2}

= \frac{\frac{4 x y - π}{2}}{2}

Apply the fraction rule:

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

= \frac{4 x y - π}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 x y - π}{4}

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

Simplify

sin (\frac{π}{4}) : \frac{2}{2}

sin (\frac{π}{4})

Use the following trivial identity:

sin (\frac{π}{4}) = \frac{2}{2}

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{2}{2}

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

Multiply fractions:

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

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

Cancel the common factor:

2

= 2 sin (\frac{4 x y - π}{4})

= - 2 - 2 sin (\frac{4 x y - π}{4})

- 2 - 2 sin (\frac{4 x y - π}{4}) = 0

Move

2

to the right side

- 2 - 2 sin (\frac{4 x y - π}{4}) = 0

Add

2

to both sides

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

Simplify

- 2 sin (\frac{4 x y - π}{4}) = 2

- 2 sin (\frac{4 x y - π}{4}) = 2

Divide both sides by

- 2

- 2 sin (\frac{4 x y - π}{4}) = 2

Divide both sides by

- 2

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

Simplify

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

Simplify

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

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

Apply the fraction rule:

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

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

Cancel the common factor:

2

= sin (\frac{4 x y - π}{4})

Simplify

\frac{2}{- 2} : - 2

\frac{2}{- 2}

Apply the fraction rule:

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

= - \frac{2}{2}

Apply radical rule:

2 = 2^{\frac{1}{2}}

= \frac{2}{2 ^{\frac{1}{2}}}

Apply exponent rule:

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

\frac{2 ^{1}}{2 ^{\frac{1}{2}}} = 2^{1 - \frac{1}{2}}

= 2^{1 - \frac{1}{2}}

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

= 2^{\frac{1}{2}}

Apply radical rule:

2^{\frac{1}{2}} = 2

= - 2

sin (\frac{4 x y - π}{4}) = - 2

sin (\frac{4 x y - π}{4}) = - 2

sin (\frac{4 x y - π}{4}) = - 2

- 1 \leq sin (x) \leq 1

No Solution for x \in R

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Frequently Asked Questions (FAQ)

What is the general solution for solvefor x,cos(xy)-sin(xy)=2 ?
The general solution for solvefor x,cos(xy)-sin(xy)=2 is No Solution for x\in\mathbb{R}