Precalculus Examples

$r = \frac{5}{1 + \cos (θ)}$

Step 1

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

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

Step 2

Multiply both sides by $r$ .

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Step 2.1

Multiply both sides by $r$ .

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Multiply $r$ by $r$ .

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

Step 2.3

Simplify the right side.

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Step 2.3.1

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

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Step 2.3.1.1

Simplify the denominator.

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Step 2.3.1.1.1

Write $1$ as a fraction with a common denominator.

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

Step 2.3.1.1.2

Combine the numerators over the common denominator.

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

Step 2.3.1.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 2.3.1.3

Rewrite using the commutative property of multiplication.

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

Step 2.3.1.4

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

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Step 2.3.1.4.1

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

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

Step 2.3.1.4.2

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

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

Step 2.3.1.4.3

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

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

Step 2.3.1.4.4

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

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

Step 2.3.1.4.5

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

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

Step 2.3.1.4.6

Add $1$ and $1$ .

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

Step 3

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

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