Precalculus Examples

$r = \frac{1}{\cos (θ) + 5 \sin (θ)}$

Step 1

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

$r = \frac{1}{\cos (θ) + 5 \frac{y}{r}}$

Step 2

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

$r = \frac{1}{\frac{x}{r} + 5 \frac{y}{r}}$

Step 3

Multiply both sides by $r$ .

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Step 3.1

Multiply both sides by $r$ .

$r \cdot r = r \frac{1}{\frac{x}{r} + 5 \frac{y}{r}}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Multiply $r$ by $r$ .

$r^{2} = r \frac{1}{\frac{x}{r} + 5 \frac{y}{r}}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify $r \frac{1}{\frac{x}{r} + 5 \frac{y}{r}}$ .

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Step 3.3.1.1

Simplify the denominator.

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Step 3.3.1.1.1

Combine $5$ and $\frac{y}{r}$ .

$r^{2} = r \frac{1}{\frac{x}{r} + \frac{5 y}{r}}$

Step 3.3.1.1.2

Combine the numerators over the common denominator.

$r^{2} = r \frac{1}{\frac{x + 5 y}{r}}$

Step 3.3.1.2

Multiply the numerator by the reciprocal of the denominator.

$r^{2} = r (1 \frac{r}{x + 5 y})$

Step 3.3.1.3

Multiply $\frac{r}{x + 5 y}$ by $1$ .

$r^{2} = r \frac{r}{x + 5 y}$

Step 3.3.1.4

Multiply $r \frac{r}{x + 5 y}$ .

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Step 3.3.1.4.1

Combine $r$ and $\frac{r}{x + 5 y}$ .

$r^{2} = \frac{r \cdot r}{x + 5 y}$

Step 3.3.1.4.2

Raise $r$ to the power of $1$ .

$r^{2} = \frac{r^{1} r}{x + 5 y}$

Step 3.3.1.4.3

Raise $r$ to the power of $1$ .

$r^{2} = \frac{r^{1} r^{1}}{x + 5 y}$

Step 3.3.1.4.4

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

$r^{2} = \frac{r^{1 + 1}}{x + 5 y}$

Step 3.3.1.4.5

Add $1$ and $1$ .

$r^{2} = \frac{r^{2}}{x + 5 y}$

Step 4

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

$x^{2} + y^{2} = \frac{x^{2} + y^{2}}{x + 5 y}$