Solve sin (x y)=- sin (x-y)

Question

Solve the equation

Solve for x

Solve for y

x = {\frac{y}{y + 1} + \frac{2 kπ}{y + 1} \frac{π - y}{y - 1} + \frac{2 kπ}{y - 1}, k \in Z

Evaluate

sin (x y) = - sin (x - y)

Rewrite the expression

sin (y x) = - sin (x - y)

Move the expression to the left side

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

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

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

Use sin (t) + sin (s) = 2 sin (\frac{t + s}{2}) cos (\frac{t - s}{2}) to transform the expression

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

Elimination the left coefficient

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

Separate the equation into 2 possible cases

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

Solve the equation

More Steps

x = \frac{y}{y + 1} + \frac{2 kπ}{y + 1}, k \in Z cos (\frac{y x - x + y}{2}) = 0

Solve the equation

More Steps

x = \frac{y}{y + 1} + \frac{2 kπ}{y + 1}, k \in Z x = \frac{π - y}{y - 1} + \frac{2 kπ}{y - 1}, k \in Z

Solution

x = {\frac{y}{y + 1} + \frac{2 kπ}{y + 1} \frac{π - y}{y - 1} + \frac{2 kπ}{y - 1}, k \in Z

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Testing for symmetry

Testing for symmetry about the origin

Testing for symmetry about the x-axis

Testing for symmetry about the y-axis

Not symmetry with respect to the origin

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Find the first derivative

Find the derivative with respect to x

Find the derivative with respect to y

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

Calculate

sin (x y) = - sin (x - y)

Take the derivative of both sides

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

Calculate the derivative

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

Calculate the derivative

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(y + x \frac{d y}{d x}) cos (x y) = - cos (x - y) (1 - \frac{d y}{d x})

Rewrite the expression

cos (x y) (y + x \frac{d y}{d x}) = - cos (x - y) (1 - \frac{d y}{d x})

Calculate

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y cos (x y) + x cos (x y) \frac{d y}{d x} = - cos (x - y) (1 - \frac{d y}{d x})

Calculate

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

Move the expression to the left side

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

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

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

Collect like terms by calculating the sum or difference of their coefficients

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

Move the constant to the right side

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

Subtract the terms

More Steps

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

Divide both sides

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

Divide the numbers

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

Solution

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

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