Solve sin(x y) = 1 / sqrt2

Question

Solve the equation

Solve for x

Solve for y

x = {\frac{π}{4 y} + \frac{2 kπ}{y} \frac{3 π}{4 y} + \frac{2 kπ}{y}, k \in Z

Evaluate

sin (x y) = \frac{1}{2}

Rewrite the expression

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

Use the inverse trigonometric function

y x = arcsin (\frac{2}{2})

Calculate

y x = \frac{π}{4} y x = \frac{3 π}{4}

Add the period of 2 kπ, k \in Z to find all solutions

y x = \frac{π}{4} + 2 kπ, k \in Z y x = \frac{3 π}{4} + 2 kπ, k \in Z

Calculate

More Steps

Evaluate

y x = \frac{π}{4} + 2 kπ

Divide both sides

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

Divide the numbers

x = \frac{\frac{π}{4} + 2 kπ}{y}

Divide the numbers

x = \frac{π}{4 y} + \frac{2 kπ}{y}

x = \frac{π}{4 y} + \frac{2 kπ}{y}, k \in Z y x = \frac{3 π}{4} + 2 kπ, k \in Z

Calculate

More Steps

Evaluate

y x = \frac{3 π}{4} + 2 kπ

Divide both sides

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

Divide the numbers

x = \frac{\frac{3 π}{4} + 2 kπ}{y}

Divide the numbers

x = \frac{3 π}{4 y} + \frac{2 kπ}{y}

x = \frac{π}{4 y} + \frac{2 kπ}{y}, k \in Z x = \frac{3 π}{4 y} + \frac{2 kπ}{y}, k \in Z

Solution

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

Symmetry with respect to the origin

Evaluate

sin (x y) = \frac{1}{2}

To test if the graph of sin (x y) = \frac{1}{2} is symmetry with respect to the origin,substitute -x for x and -y for y

sin (- x (- y)) = \frac{1}{2}

Multiplying or dividing an even number of negative terms equals a positive

sin (x y) = \frac{1}{2}

Solution

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}{x}

Calculate

sin (x y) = \frac{1}{2}

Take the derivative of both sides

\frac{d}{d x} (sin (x y)) = \frac{d}{d x} (\frac{1}{2})

Calculate the derivative

(y + x \frac{d y}{d x}) cos (x y) = \frac{d}{d x} (\frac{1}{2})

Calculate the derivative

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

Rewrite the expression

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

Rewrite the expression

y + x \frac{d y}{d x} = 0

Move the constant to the right side

x \frac{d y}{d x} = 0 - y

Removing 0 doesn't change the value,so remove it from the expression

x \frac{d y}{d x} = - y

Divide both sides

\frac{x \frac{d y}{d x}}{x} = \frac{- y}{x}

Divide the numbers

\frac{d y}{d x} = \frac{- y}{x}

Solution

\frac{d y}{d x} = - \frac{y}{x}

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Find the second derivative

Find the second derivative with respect to x

Find the second derivative with respect to y

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

Calculate

sin (x y) = \frac{1}{2}

Take the derivative of both sides

\frac{d}{d x} (sin (x y)) = \frac{d}{d x} (\frac{1}{2})

Calculate the derivative

(y + x \frac{d y}{d x}) cos (x y) = \frac{d}{d x} (\frac{1}{2})

Calculate the derivative

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

Rewrite the expression

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

Rewrite the expression

y + x \frac{d y}{d x} = 0

Move the constant to the right side

x \frac{d y}{d x} = 0 - y

Removing 0 doesn't change the value,so remove it from the expression

x \frac{d y}{d x} = - y

Divide both sides

\frac{x \frac{d y}{d x}}{x} = \frac{- y}{x}

Divide the numbers

\frac{d y}{d x} = \frac{- y}{x}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

\frac{d y}{d x} = - \frac{y}{x}

Take the derivative of both sides

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

Calculate the derivative

\frac{d ^{2} y}{d x ^{2}} = \frac{d}{d x} (- \frac{y}{x})

Use differentiation rules

\frac{d ^{2} y}{d x ^{2}} = - \frac{\frac{d}{d x} ( y ) \times x - y \times \frac{d}{d x} ( x )}{x ^{2}}

Calculate the derivative

More Steps

Evaluate

\frac{d}{d x} (y)

Use differentiation rules

\frac{d}{d y} (y) \times \frac{d y}{d x}

Use \frac{d}{d x} x^{n} = n x^{n - 1} to find derivative

\frac{d y}{d x}

\frac{d ^{2} y}{d x ^{2}} = - \frac{\frac{d y}{d x} \times x - y \times \frac{d}{d x} ( x )}{x ^{2}}

Use \frac{d}{d x} x^{n} = n x^{n - 1} to find derivative

\frac{d ^{2} y}{d x ^{2}} = - \frac{\frac{d y}{d x} \times x - y \times 1}{x ^{2}}

Use the commutative property to reorder the terms

\frac{d ^{2} y}{d x ^{2}} = - \frac{x \frac{d y}{d x} - y \times 1}{x ^{2}}

Any expression multiplied by 1 remains the same

\frac{d ^{2} y}{d x ^{2}} = - \frac{x \frac{d y}{d x} - y}{x ^{2}}

Use equation \frac{d y}{d x} = - \frac{y}{x} to substitute

\frac{d ^{2} y}{d x ^{2}} = - \frac{x ( - \frac{y}{x} ) - y}{x ^{2}}

Solution

More Steps

Calculate

- \frac{x ( - \frac{y}{x} ) - y}{x ^{2}}

Multiply the terms

More Steps

Evaluate

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

Multiplying or dividing an odd number of negative terms equals a negative

- x \times \frac{y}{x}

Cancel out the common factor x

- 1 \times y

Multiply the terms

- y

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

Subtract the terms

More Steps

Simplify

- y - y

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

(- 1 - 1) y

Subtract the numbers

- 2 y

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

Divide the terms

- (- \frac{2 y}{x ^{2}})

Calculate

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

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

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