# How do you convert r = 1 + cos (theta) into rectangular form?

Jul 5, 2017

${x}^{2} + {y}^{2} = \sqrt{{x}^{2} + {y}^{2}} + x$

#### Explanation:

Converting from polar to rectangular form:

$x = r \cos \left(\theta\right)$, $y = r \sin \left(\theta\right)$

${x}^{2} + {y}^{2} = {r}^{2} \to r = \sqrt{{x}^{2} + {y}^{2}}$

Here our polar equation is: $r = 1 + \cos \left(\theta\right)$

Multiply both sides by $r \to {r}^{2} = r \left(1 + \cos \left(\theta\right)\right)$

$\therefore {r}^{2} = r + r \cos \left(\theta\right)$

Substituting for $r , {r}^{2} \mathmr{and} r \cos \left(\theta\right)$ yields:

${x}^{2} + {y}^{2} = \sqrt{{x}^{2} + {y}^{2}} + x$

Which is our polar equation in rectangular form.

NB: This is the equation of the cardioid below.

graph{ x^2+y^2 =sqrt(x^2+y^2) + x [-1.835, 4.325, -1.478, 1.602]}