Calculus Examples

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

Step 1

Rewrite the equation.

$\cos (y) d y = \sin (x) d x$

Step 2

Integrate both sides.

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Step 2.1

Set up an integral on each side.

$\int \cos (y) d y = \int \sin (x) d x$

Step 2.2

The integral of $\cos (y)$ with respect to $y$ is $\sin (y)$ .

$\sin (y) + C_{1} = \int \sin (x) d x$

Step 2.3

The integral of $\sin (x)$ with respect to $x$ is $- \cos (x)$ .

$\sin (y) + C_{1} = - \cos (x) + C_{2}$

Step 2.4

Group the constant of integration on the right side as $K$ .

$\sin (y) = - \cos (x) + K$

Step 3

Solve for $y$ .

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Step 3.1

Rewrite the equation as $- \cos (x) + K = \sin (y)$ .

$- \cos (x) + K = \sin (y)$

Step 3.2

Subtract $K$ from both sides of the equation.

$- \cos (x) = \sin (y) - K$

Step 3.3

Divide each term in $- \cos (x) = \sin (y) - K$ by $- 1$ and simplify.

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Step 3.3.1

Divide each term in $- \cos (x) = \sin (y) - K$ by $- 1$ .

$\frac{- \cos (x)}{- 1} = \frac{\sin (y)}{- 1} + \frac{- K}{- 1}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Dividing two negative values results in a positive value.

$\frac{\cos (x)}{1} = \frac{\sin (y)}{- 1} + \frac{- K}{- 1}$

Step 3.3.2.2

Divide $\cos (x)$ by $1$ .

$\cos (x) = \frac{\sin (y)}{- 1} + \frac{- K}{- 1}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Simplify each term.

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Step 3.3.3.1.1

Move the negative one from the denominator of $\frac{\sin (y)}{- 1}$ .

$\cos (x) = - 1 \cdot \sin (y) + \frac{- K}{- 1}$

Step 3.3.3.1.2

Rewrite $- 1 \cdot \sin (y)$ as $- \sin (y)$ .

$\cos (x) = - \sin (y) + \frac{- K}{- 1}$

Step 3.3.3.1.3

Dividing two negative values results in a positive value.

$\cos (x) = - \sin (y) + \frac{K}{1}$

Step 3.3.3.1.4

Divide $K$ by $1$ .

$\cos (x) = - \sin (y) + K$

Step 3.4

Take the inverse cosine of both sides of the equation to extract $y$ from inside the cosine.

$x = \arccos (- \sin (y) + K)$

Step 3.5

Rewrite the equation as $\arccos (- \sin (y) + K) = x$ .

$\arccos (- \sin (y) + K) = x$

Step 3.6

Take the inverse arccosine of both sides of the equation to extract $\sin (y)$ from inside the arccosine.

$- \sin (y) + K = \cos (x)$

Step 3.7

Subtract $K$ from both sides of the equation.

$- \sin (y) = \cos (x) - K$

Step 3.8

Divide each term in $- \sin (y) = \cos (x) - K$ by $- 1$ and simplify.

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Step 3.8.1

Divide each term in $- \sin (y) = \cos (x) - K$ by $- 1$ .

$\frac{- \sin (y)}{- 1} = \frac{\cos (x)}{- 1} + \frac{- K}{- 1}$

Step 3.8.2

Simplify the left side.

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Step 3.8.2.1

Dividing two negative values results in a positive value.

$\frac{\sin (y)}{1} = \frac{\cos (x)}{- 1} + \frac{- K}{- 1}$

Step 3.8.2.2

Divide $\sin (y)$ by $1$ .

$\sin (y) = \frac{\cos (x)}{- 1} + \frac{- K}{- 1}$

Step 3.8.3

Simplify the right side.

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Step 3.8.3.1

Simplify each term.

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Step 3.8.3.1.1

Move the negative one from the denominator of $\frac{\cos (x)}{- 1}$ .

$\sin (y) = - 1 \cdot \cos (x) + \frac{- K}{- 1}$

Step 3.8.3.1.2

Rewrite $- 1 \cdot \cos (x)$ as $- \cos (x)$ .

$\sin (y) = - \cos (x) + \frac{- K}{- 1}$

Step 3.8.3.1.3

Dividing two negative values results in a positive value.

$\sin (y) = - \cos (x) + \frac{K}{1}$

Step 3.8.3.1.4

Divide $K$ by $1$ .

$\sin (y) = - \cos (x) + K$

Step 3.9

Take the inverse sine of both sides of the equation to extract $y$ from inside the sine.

$y = \arcsin (- \cos (x) + K)$

Step 3.10

Since $y$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\arccos (- \sin (y) + K) = x$

Step 3.11

Take the inverse arccosine of both sides of the equation to extract $\sin (y)$ from inside the arccosine.

$- \sin (y) + K = \cos (x)$

Step 3.12

Subtract $K$ from both sides of the equation.

$- \sin (y) = \cos (x) - K$

Step 3.13

Divide each term in $- \sin (y) = \cos (x) - K$ by $- 1$ and simplify.

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Step 3.13.1

Divide each term in $- \sin (y) = \cos (x) - K$ by $- 1$ .

$\frac{- \sin (y)}{- 1} = \frac{\cos (x)}{- 1} + \frac{- K}{- 1}$

Step 3.13.2

Simplify the left side.

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Step 3.13.2.1

Dividing two negative values results in a positive value.

$\frac{\sin (y)}{1} = \frac{\cos (x)}{- 1} + \frac{- K}{- 1}$

Step 3.13.2.2

Divide $\sin (y)$ by $1$ .

$\sin (y) = \frac{\cos (x)}{- 1} + \frac{- K}{- 1}$

Step 3.13.3

Simplify the right side.

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Step 3.13.3.1

Simplify each term.

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Step 3.13.3.1.1

Move the negative one from the denominator of $\frac{\cos (x)}{- 1}$ .

$\sin (y) = - 1 \cdot \cos (x) + \frac{- K}{- 1}$

Step 3.13.3.1.2

Rewrite $- 1 \cdot \cos (x)$ as $- \cos (x)$ .

$\sin (y) = - \cos (x) + \frac{- K}{- 1}$

Step 3.13.3.1.3

Dividing two negative values results in a positive value.

$\sin (y) = - \cos (x) + \frac{K}{1}$

Step 3.13.3.1.4

Divide $K$ by $1$ .

$\sin (y) = - \cos (x) + K$

Step 3.14

Take the inverse sine of both sides of the equation to extract $y$ from inside the sine.

$y = \arcsin (- \cos (x) + K)$