# How do you graph the equation r = 1 + cos( theta )?

Jul 15, 2016

Graph of ${x}^{2} + {y}^{2} = \sqrt{{x}^{2} + {y}^{2}} + x$ or $r = 1 + \cos \left(\theta\right)$
graph{x^2+y^2=sqrt(x^2+y^2)+x [-10, 10, -5, 5]}

#### Explanation:

In case you are trying to graph the equation in rectangular form, here's a way to get it to rectangular form and graph it.

We can make use of the following formulas when trying to convert from polar to rectangular:

$x = r \cos \left(\theta\right)$ and $y = r \sin \left(\theta\right)$
${r}^{2} = {x}^{2} + {y}^{2}$

Now we can rewrite our equation:

$r = 1 + \cos \left(\theta\right)$

Multiplying both sides by $r$ gives us

${r}^{2} = r \left(1 + \cos \left(\theta\right)\right)$

$= r + r \cos \left(\theta\right)$

Substituting the value of $r = \sqrt{{x}^{2} + {y}^{2}}$ into our equation yields

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

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

So our equation becomes

${x}^{2} + {y}^{2} = \sqrt{{x}^{2} + {y}^{2}} + x$, which is equivalent to $r = 1 + \cos \left(\theta\right)$.

Below are a few graphs.

Graph of ${x}^{2} + {y}^{2} = \sqrt{{x}^{2} + {y}^{2}} + x$ or $r = 1 + \cos \left(\theta\right)$

graph{x^2+y^2=sqrt(x^2+y^2)+x [-10, 10, -5, 5]}