Calculus Examples

$r = 4 \cot (θ) \csc (θ)$

Step 1

Simplify the right side.

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Step 1.1

Simplify $4 \cot (θ) \csc (θ)$ .

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Step 1.1.1

Rewrite $\cot (θ)$ in terms of sines and cosines.

$r = 4 \frac{\cos (θ)}{\sin (θ)} \csc (θ)$

Step 1.1.2

Combine $4$ and $\frac{\cos (θ)}{\sin (θ)}$ .

$r = \frac{4 \cos (θ)}{\sin (θ)} \csc (θ)$

Step 1.1.3

Rewrite $\csc (θ)$ in terms of sines and cosines.

$r = \frac{4 \cos (θ)}{\sin (θ)} \cdot \frac{1}{\sin (θ)}$

Step 1.1.4

Multiply $\frac{4 \cos (θ)}{\sin (θ)} \cdot \frac{1}{\sin (θ)}$ .

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Step 1.1.4.1

Multiply $\frac{4 \cos (θ)}{\sin (θ)}$ by $\frac{1}{\sin (θ)}$ .

$r = \frac{4 \cos (θ)}{\sin (θ) \sin (θ)}$

Step 1.1.4.2

Raise $\sin (θ)$ to the power of $1$ .

$r = \frac{4 \cos (θ)}{\sin^{1} (θ) \sin (θ)}$

Step 1.1.4.3

Raise $\sin (θ)$ to the power of $1$ .

$r = \frac{4 \cos (θ)}{\sin^{1} (θ) \sin^{1} (θ)}$

Step 1.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$r = \frac{4 \cos (θ)}{{\sin (θ)}^{1 + 1}}$

Step 1.1.4.5

Add $1$ and $1$ .

$r = \frac{4 \cos (θ)}{\sin^{2} (θ)}$

Step 2

Since $\sin (θ) = \frac{y}{r}$ , replace $\sin (θ)$ with $\frac{y}{r}$ .

$r = \frac{4 \cos (θ)}{{(\frac{y}{r})}^{2}}$

Step 3

Since $\cos (θ) = \frac{x}{r}$ , replace $\cos (θ)$ with $\frac{x}{r}$ .

$r = \frac{4 \frac{x}{r}}{{(\frac{y}{r})}^{2}}$

Step 4

Multiply both sides by $r$ .

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Step 4.1

Multiply both sides by $r$ .

$r \cdot r = r \frac{4 \frac{x}{r}}{{(\frac{y}{r})}^{2}}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Multiply $r$ by $r$ .

$r^{2} = r \frac{4 \frac{x}{r}}{{(\frac{y}{r})}^{2}}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Simplify $r \frac{4 \frac{x}{r}}{{(\frac{y}{r})}^{2}}$ .

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Step 4.3.1.1

Combine $4$ and $\frac{x}{r}$ .

$r^{2} = r \frac{\frac{4 x}{r}}{{(\frac{y}{r})}^{2}}$

Step 4.3.1.2

Apply the product rule to $\frac{y}{r}$ .

$r^{2} = r \frac{\frac{4 x}{r}}{\frac{y^{2}}{r^{2}}}$

Step 4.3.1.3

Multiply the numerator by the reciprocal of the denominator.

$r^{2} = r (\frac{4 x}{r} \cdot \frac{r^{2}}{y^{2}})$

Step 4.3.1.4

Combine.

$r^{2} = r \frac{4 x r^{2}}{r y^{2}}$

Step 4.3.1.5

Cancel the common factor of $r$ .

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Step 4.3.1.5.1

Factor $r$ out of $r y^{2}$ .

$r^{2} = r \frac{4 x r^{2}}{r (y^{2})}$

Step 4.3.1.5.2

Cancel the common factor.

$r^{2} = r \frac{4 x r^{2}}{r y^{2}}$

Step 4.3.1.5.3

Rewrite the expression.

$r^{2} = \frac{4 x r^{2}}{y^{2}}$

Step 5

Since $r^{2} = x^{2} + y^{2}$ , replace $r^{2}$ with $x^{2} + y^{2}$ and $r$ with $\sqrt{x^{2} + y^{2}}$ .

$x^{2} + y^{2} = \frac{4 x (x^{2} + y^{2})}{y^{2}}$