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.

Now on Google+! Add me to your circles!

I've finally gotten around to connecting my site to Google+!

I've set up a page specifically for this Math Concepts Explained blog.  Visit my Google+ page at https://plus.google.com/114879190180011260820 and add me to your circles!  I'm planning on posting brief bits of my blog posts there as I generate new ones, but you're going to have to visit this blog for the full deal!  You can also send me emails to ask questions or leave comments, if you don't want to do so on the blog's comment section.  Please send your math questions or comments to my new email address at mathconcepts101 [at] gmail [dot] com.

I'm also planning on starting a Math Concepts Explained profile on Facebook soon!  (Update:  My Facebook page is now live!  Come visit, click the Like button, and add me to your Friends lists!)

And for those of you who are new to my site, I also have a Twitter account already set up.  My profile is MathConcepts on Twitter (click the Follow me button near the top of the page).  You can easily connect with me on there as well!

Hopefully these new additions are going to be big for the blog!  Thanks to everyone for visiting!

Degree of a Polynomial

One of the most common questions you will find when first learning about polynomials, "How do you find the degree of a polynomial," sounds incredibly complicated and difficult, but really couldn't be simpler.  Sometimes, you don't need to use a bunch of math concepts to find the answer, or to do several mathematical manipulations to arrive at the solution.  Often, you don't have to do anything but just LOOK at the equation, and you can figure out the degree of a polynomial.  And that is because finding the degree of a polynomial is simple.

The degree of a polynomial is simply the highest exponent of the variable in the equation.

Look at this question, and you will see just how easy it can be:

What is the degree of the polynomial:  x³ + x = 5.

Yes, it is as simple as it sounds.  The highest power of x in this polynomial is 3, and so that is the degree.  See? Told you it can be easy!

Your questions will likely look similar to that one.  However, there are a few things to say as well, just in case you get a curve ball question.

Non-zero terms (e.g. integers) have a variable with an exponent of 0 (which means that the variable actually equals 1, and therefore you don't need to show it).  If you ONLY have non-zero integer, and therefore assume you have a variable with an exponent of 0, you can say that the degree of this non-zero constant polynomial is 0.  That is, constants are zero-degree.  However, that is only for non-zero polynomials.  Zero itself has an undefined degree.

So, the degree of the non-zero constant polynomial 8 is 0.  Easy.

Of course, you will inevitably get harder questions that do actually require a bit of work to arrive at your solution, but that will only be more of the same kind of mathematical concepts you've been studying, such as how to FOIL polynomials, or factoring by grouping.  In the end, when you have your polynomial expression, all you need to do is determine what is the highest exponent of the variable, and then state that as the degree.  Try some practice questions, and you will find that you won't be asking "how do you find the degree of a polynomial" anymore!

One last thought I'd like you to consider:  I have seen people searching my site for help with things like "degree polynomial" or "degree polynomials" and I think that sounds very strange, and suggests to me that this math concept is not being taught very well at all in the classroom.  Technically, you don't have "degree polynomials" per se.  ALL polynomials, except zero, have a degree which can be anything from third degree polynomials to higher degree polynomials such as ten-thousand degree polynomials (though I certainly don't want to be the one working with a polynomial of that degree!).  If you think about it, to say something like a "degree polynomial" is to be redundant!

In any case, if you arrived here after searching for help with finding the degree of a polynomial, or for "degree polynomial," I hope this post has been informative and helpful for you.  It's not a math concept that is as hard as it sounds, so hopefully my post makes sense and you can take something away to help teach other students who don't know what "degree polynomials" are.  :-)  As always, please remember to +1 me below if you liked my post!  Also follow me on Twitter with the button above!

Use Siri for Math on iPhone 4S

For people who recently bought Apple's new superphone, the iPhone 4S, one of the most compelling new features is the "virtual assistant" Siri.  It's a groundbreaking new app that does a pretty good job of understanding natural language spoken to it through the iPhone's mic, and then responding to you with some kind of acknowledgement or action.  Some of the cool things that Siri can do are highlighted everywhere in the news: send a text message to someone, call someone, setup an appointment or reminder.  In some countries (for now, only a few), Siri can even provide you with directions to a location, or understand things like "i'm hungry" and respond to you with nearby restaurants.  However, as useful as these functions are, one of Siri's less advertised features is an interaction with the knowledge engine Wolfram Alpha.

Wolfram Alpha works similarly to Google, but rather than returning a list of search results to a query, it actually attempts to answer your question with data in its gigantic fact database.  So, if you ask Wolfram Alpha "what is the fifth planet from the Sun" or "what is the capital city of Belgium," it will respond with "Jupiter" and "Brussels" as well as a boatload of information specifically about those entries.  Wolfram Alpha is truly amazing.  Now, if you put this together with Siri on the iPhone 4S, what you are able to do is simply ask Siri any factual question (without having to go into a browser or type out your question), it will query Wolfram Alpha for you, and then return the information you're looking for.

So, what does all of this have to do with math?  How does this allow you to use Siri for math on iPhone 4S?

Well, if you input math questions into Wolfram Alpha, it will come back with the answer.  This works especially well for unit conversions, or looking up constants, or working with math expressions.  Having the ability to simply ask Siri your math question, and have the app return the calculation for you is a remarkable shortcut to writing out and working through complex calculatons.  You don't even have to punch numbers into a calculator.  All you have to do is ask Siri your math questions, and it will return the solutions.

Of course this isn't going to TEACH you how to perform these calculations, but when you know about these math concepts already, this is an absolutely fantastic math shortcut that means you don't have to do the hard work!  Check out some of the ways that you can use Siri for math on iPhone 4S.  These are just a few examples, and I'm sure you can find even more.  (And remember that Wolfram Alpha can handle all kinds of factual questions, so you're not restricted to asking Siri only math questions!)  Once you realize how powerful Siri is, you will find that it can be a very powerful, helpful, and convenient calculator!

Show you the value of mathematical constants:

Calculate percentages:

Calculate square roots:

 Calculate reciprocals:

Perform unit conversions:

Perform arithmetic on a string of integers:

Perform arithmetic on fractions:

Perform complex calculations:

If you notice in my pictures, Siri doesn't just respond with the answer, but also with a host of other data such as unit conversions or different presentations of data.  It will also always show you what it understood for the original question, so you can verify that it understood you correctly and actually performed the computations on the proper input.

Now, admittedly, Siri is not going to help you learn math any easier or faster.  You still need to put in time with the usual techniques... i.e. you still have to do your homework and study!  And, you won't be allowed to take your iPhone 4S and Siri with you into an exam any time soon.  Just imagine everyone in an exam room talking to their iPhones and all the Siri's talking back!  However, when you get to a point in your studies that you understand the math concepts being taught, and you've done plenty of practice to understand how to arrive at the correct answer, then having the ability to use Siri for math on iPhone 4S to quickly and accurately find what you need to know is incredibly valuable!  (Please hit the Google +1 below if you found this article useful!)