# How do you find the cartesian graph of r = 5sin(θ)?

Nov 27, 2015

You basically get a circle.

#### Explanation:

Consider the following diagram:

We can see that the relationships between rectangular and polar coordinates are:
$r = \sqrt{{x}^{2} + {y}^{2}}$
$\theta = \arctan \left(\frac{y}{x}\right)$
and:
$x = r \cos \left(\theta\right)$
$y = r \sin \left(\theta\right)$

Given our expression:
$r = 5 \sin \left(\theta\right)$

multiply by $r$ both sides:
${r}^{2} = 5 r \sin \left(\theta\right)$
so that you get, using our relationships of conversion:

$\textcolor{red}{{x}^{2} + {y}^{2} = 5 y}$ (1)

which is the equation of a circle centered at $\left({x}_{c} = 0 , {y}_{c} = \frac{5}{2}\right)$ and with radius $r = \frac{5}{2}$, whose equation is found from the general form of a circle:
${\left(x - {x}_{c}\right)}^{2} + {\left(y - {y}_{c}\right)}^{2} = {r}^{2}$
(try to substitute the values of the center and radius and you'll find (1)).
You can now plot it directly.

Next, to have some fun, I used Excel to evaluate, using our relationships of conversion, the coordinates $x$ and $y$ (in red) and plot them: