Convert Polar to Rectangular

In this post, I am going to describe the theory that allows you to convert polar to rectangular coordinates.  This follows up my previous post that describes graphing polar coordinates, where I introduced the concept of this new method of graphing.  However, the relationships that I will show you here will hopefully allow you to see the connection between polar and Cartesian coordinate systems, which will make them easier for you to work with!

To quickly refresh what I explained last time, start at the origin (or pole) of your graph and extend the polar axis line out to the right.  This is your reference line that will help you describe the location of any other point.  Now, pick a point P somewhere, and this is described by its distance (radius), r, from the origin (if you were to connect it to there) and the angle, ɵ, by which that line has rotated away from the polar axis.  Whereas points in the rectangular coordinate system are described as P (x, y), points in the polar coordinate system are described as P (r, ɵ).

Now, let's take a closer look at the relationship between P (x, y) and P (r, ɵ).  This will use a little bit of trigonometry, so review it in my posts about sine, cosine, and tangent if you need to brush up!

To do this, let's superimpose the two coordinate systems, meaning that we will assume that the origins of each are in the same place, and the polar axis is the same as the positive x-axis.  You should be able to see, with the help of the following handy figure, that our point P can be described by P (x, y) and P (r, ɵ).  You can also see that, by combining the two coordinate systems, we have formed a triangle which has sides of lengths x and y and hypotenuse r.  This triangle will be the basis that allows us to convert polar to rectangular coordinates.

Now, if we apply our rules and identities of trigonometry, the relationships of the triangle sides and angles will connect the polar coordinate system and rectangular coordinate system.  And with these relationships, you will be able to convert from polar to rectangular, and also back again to convert rectangular to polar.

The relationships are quite simply the basic trigonometry identities that you already know:

sin ɵ = opposite / hypotenuse

cos ɵ = adjacent / hypotenuse

tan ɵ = opposite / adjacent

Now, if we substitute in the names for our sides, and rearrange to have polar terms on one side and Cartesian terms on the other, we arrive at the following relationships:

sin ɵ = y / r  -----> y = r sin ɵ 

cos ɵ = x / r -----> x = r cos ɵ 

tan ɵ = y / x

Furthermore, we can apply the Theorem of Pythagoras to give us another useful relationship:

r² = x² + y²

And there you have it!  Easily derived connections between polar and rectangular coordinate systems.  You will find as you work through you studies that sometimes some expressions may be easier to work with in one coordinate system or the other.  Keep this in mind, especially when you begin to work with polar equations.  Using these quick methods to convert polar to rectangular coordinates may help you to get through your problems a lot faster (and easier)!

In my next post, I'll show you some more things about polar coordinates!