# How do you sketch the graph of the polar equation and find the tangents at the pole of r=3sintheta?

Nov 17, 2017

$\theta = 0$

#### Explanation:

The relation between polar coordinates $\left(r , \theta\right)$ and Cartesian coordinates $\left(x , y\right)$ are $x = r \cos \theta$, $y = r \sin \theta$ and hence ${r}^{2} = {x}^{2} + {y}^{2}$.

The equation $r = 3 \sin \theta$ can be written as ${r}^{2} = 3 r \sin \theta$

i.e. ${x}^{2} + {y}^{2} = 3 y$ which can be modified as

${x}^{2} + {\left(y - \frac{3}{2}\right)}^{2} = {\left(\frac{3}{2}\right)}^{2}$

which is equation of a circle with center at $\left(0 , \frac{3}{2}\right)$ and radius $\frac{3}{2}$. Its graph is as shown below.

graph{x^2+y^2=3y [-5, 5, -1, 4]}

and tangent at pole i.e. $\left(0 , 0\right)$ is $y = 0$,

which in polar form is $\theta = 0$