Announcing... the Mathematics and Multimedia Carnival 21!

I am pleased to announce that the 21st edition of the Mathematics and Multimedia Carnival will be hosted here on Math Concepts Explained in March!

For those who are new to blog carnivals (like myself!), Guillermo has put together a nice summary on his website that explains what is a blog carnival.  Quite simply, it is just a collection of related blogs or websites that are submitted by readers and then have been summarized and listed into a single blog entry by the carnival host.  A carnival may be compared to a magazine, in that various writers submit their articles to be published, and then the magazine editor reviews all the entries and puts them together into a final, polished magazine.

As I mentioned, I have never hosted a carnival, nor have I participated in one, so I admit to being very unfamiliar with the process surrounding this.  However, I am looking forward to it as an opportunity to make some new friends and learn some new things.  So, please bear with me as I figure this out as I go!

With that, look for an official call for entries in the coming days.  I will explain the submission process at that time.  :)

In the meantime, I am happy to direct you to Guillermo's website GeoGebra Applet Central, where he is hosting the 20th edition of the Mathematics and Multimedia Carnival later in February.

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!

Graphing - Polar Coordinates

Polar coordinates are a system of describing points on a plane, in a way that is similar yet still quite different from what you have previously used.  I am going to explain what polar coordinates are here, and then in a subsequent post, I am going to explain their relationship to the graphing system that you already know about.  Hopefully you will find that graphing polar coordinates are not really any harder than what you already know how to do!

Up until now, you have likely only ever worked with rectangular coordinates, otherwise known as the Cartesian coordinate system.  This is the familiar horizontal x-axis and vertical y-axis, which intersect at the origin and separate the coordinate plane into four quadrants.  Any point on the xy plane can be described by it's location relative to these axes, which requires knowing how far horizontal and how far vertical it is away from the origin.  These are the x-coordinate and y-coordinate of the point, and together they form an ordered pair, which completely describes the point's location on the xy plane.  (For those looking for extra trivia: the x-coordinate has the technical name "abscissa" while the y-coordinate is called the "ordinate."  Try explaining that to your teacher for bonus points!  Also of interest, this Cartesian coordinate system is named after René Descartes, who was a French philosopher and mathematician whose work, among other things, first described the connection between algebra and geometry.)

Polar form is quite different from this rectangular coordinate system.  It requires a different way of thinking about the points on the graph.  Don't think of an x-y axis system for now.  To start, just place a point, and then from that point extend a line out to the right (this is conventionally what is done, though technically you could draw it in any direction).  The starting point, O, is the origin of this coordinate system, but in this case it may also rightfully be called the pole.  The line that you have extended from the pole is the polar axis.  We will define points on the plane relative to this axis (as opposed to the Cartesian system, which defines points relative to two axes).

So then, now we need a point P to talk about.  Pick any point, and then draw a line (called r, as in radius) that connects it to the pole.  The basic premise of a polar coordinate graph is that you describe the point by providing its distance from the pole, and also the angle (ɵ, theta) by which r is rotated away from the polar axis.  Knowing these two dimensions is enough information to fully describe where your point is located on the plane.  So, while a Cartesian ordered pair looks like (x, y), a polar ordered pair looks like (r, ɵ).  So, r and ɵ are referred to as polar coordinates of P, and it is written as P(r, ɵ) which explains that P is the point with polar coordinates (r, ɵ).

To assist you in drawing your polar graphs, polar coordinate graph paper is available.  Instead of being labeled as x and y axes, they are labeled with angles (in radians... I'll cover radians in another post as well).  However, I don't think you really need to use it until you start graphing more in-depth polar equations.  Whenever I draw a polar graph, I just draw my own.  :)  Briefly, 180°= π radians, 90° = π/2 radians, 270° = 3π/2 radians, and 360° = 2π radians.  I'm not going to go into anymore detail than this about radians, because if you're at the point where you're studying polar coordinate graphs, you've already covered radians.  :)

Using this new coordinate system and working with polar equations will allow you to draw some really cool polar graphs.  Expect to see plenty of "four-leaf clover" graphs and "spiral" graphs, among other really interesting designs.  Just try to think of your graph as a series of points along a line that is rotating or sweeping around the pole in the center, moving closer or further as it revolves around it, as described by its equation.  I will explain some polar equations in a future post that will highlight some common and cool polar graphs.

In my next post, I want to explain to you how to convert from polar coordinates to rectangular coordinates.  With a little bit of logic, you will see that the derivation to connect polar form to rectangular form is quite easy.  Similarly, going from rectangular coordinates to polar coordinates follows along the same theory.  This connection is a helpful basis to build upon, since by this point you are already very familiar with the Cartesian coordinate system.  Layering the polar coordinate system on top is the best way to learn this new approach to graphing.