# How do you find the polar coordinates of the point?

Mar 14, 2018

I am assuming that you want to convert a point in Cartesian Coordinate form to Polar Coordinate Form.

I will use an example in the explanation below.

#### Explanation:

Let us say that we want to convert a point $\left(0 , 2\right)$ in Cartesian Form to Polar Form.

Let us analyze the image below:

We have a right-triangle and hence we can use Pythagoras Theorem to define a relationship:

${x}^{2} + {y}^{2} = {r}^{2}$

$\tan \theta = \frac{y}{x}$

We can see that

$\cos \theta = \frac{x}{r}$

Hence,

color(blue)(x=r cos theta color(red)([ 1 ]

$\sin \theta = \frac{y}{r}$

Hence,

color(blue)(y = r sin theta color(red)([ 2 ]

Observe that, if we are given the point $\left(x , y\right)$, then we can define $\left(r , \theta\right)$ and express $\left(x , y\right)$ in terms of $\left(r \cos \theta , r \sin \theta\right)$.

Now, let us work on the point in Cartesian coordinate form, in our example:

$\left(0 , 2\right)$

Let us plot this point and look at the graph:

Now, we are in a position to convert the $\left(0 , 2\right)$ to equivalent Polar form using color(red)([ 1 ] and color(red)([ 2 ]

Polar form is color(green)((r, theta)

rArr color(blue)((2, pi/2)

Hope it helps.