Convert to Rectangular r=2/(1+sin(theta))

$r = \frac{2}{1 + \sin (θ)}$

Step 1

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

$r = \frac{2}{1 + \frac{y}{r}}$

Step 2

Multiply both sides by $r$ .

$r \cdot r = r \frac{2}{1 + \frac{y}{r}}$

Simplify the left side.

Multiply $r$ by $r$ .

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

Simplify the right side.

Simplify $r \frac{2}{1 + \frac{y}{r}}$ .

Simplify the denominator.

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

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

Combine the numerators over the common denominator.

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

Multiply the numerator by the reciprocal of the denominator.

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

Rewrite using the commutative property of multiplication.

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

Multiply $2 r \frac{r}{r + y}$ .

Combine $2$ and $\frac{r}{r + y}$ .

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

Combine $r$ and $\frac{2 r}{r + y}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Step 3

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

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