Pi

I want to eventually talk about what is commonly known as the most beautiful equation in math, but first you have to understand a lot of other stuff. I need to talk about a bunch of topics before it will make actual sense, so right now I will talk about the number pi or $\pi$ which everybody has seen before, but not many people understand.

First you need to know what a circle is. You probably know what a circle looks like, but the best way to describe it in words is to say that, “A circle is the set of all points that are a given distance away from some point.” So, you pick some random point on a paper, you pick a distance which we call the radius of the circle, and then you find all possible points that are that distance away from the initial point. Doing this will give you a circle.

Now we have a circle. Just by doing this we have some important things that we can look at. The first is the radius, which is the distance from the center to any point on the circle. Then we have the diameter, which is twice the radius or is the length of a line between two points on the circle that goes through the center. The last piece of important information is the circumference, which is the length of the outline of the circle.

A long time ago in a place far away people tried to divide the circumference by the diameter and discovered that no matter what circle you have, this ratio will always be the same. Then they called this ratio $\pi \text{ which is roughly 3.14} \ldots$
It is an irrational number so the digits will never end.

Because of this definition, $\pi$ is also used as a 180 degree rotation. Instead of saying that something is rotated by 180 degrees, math people say it is rotated by $\pi$. Similarly, math people say that something is rotated $2\pi$ instead of 360 degrees, or $\frac {1}{2} \pi$
instead of 90 degrees and so on. This is because a 180 degree rotation in a circle is like going from one end of a diameter line to the other end.

Next, the trig functions would come in which usually scares everybody away. Trig functions just take an angle in and spit out a ratio based on that angle. They are only scary because many teachers force students to memorize instead of think.

To show how they work, we like to use the most basic circle (called the unit circle) instead of an arbitrary circle, but I actually think that the arbitrary circle is more intuitive. So consider this arbitrary circle with radius “r.” From the center I can draw any radius I want, below are some examples.
$\setlength{\unitlength}{1mm} \begin{picture}(140, 40) \put(0,20){\circle{14}} \put(0,20){\line(1,0){7}} \put(0.7,20.3){r} \put(15,20){\circle{14}} \put(15,20){\line(1,1){5}} \put(15.7,24){r} \put(30,20){\circle{14}} \put(30,20){\line(0,1){7}} \put(30.7,24){r} \put(45,20){\circle{14}} \put(45,20){\line(-1,1){5}} \put(44.7,24){r} \put(60,20){\circle{14}} \put(60,20){\line(-1,0){7}} \put(60.3,20.3){r} \put(75,20){\circle{14}} \put(75,20){\line(-1,-1){5}} \put(74.7,16){r} \put(90,20){\circle{14}} \put(90,20){\line(0,-1){7}} \put(90.7,16){r} \put(105,20){\circle{14}} \put(105,20){\line(1,-1){5}} \put(104.7,16){r} \put(120,20){\circle{14}} \put(120,20){\line(1,0){7}} \put(120.3,20.3){r} \end{picture}$

For any given radial line, I can draw a right triangle:
$\setlength{\unitlength}{1mm} \begin{picture}(140, 40) \put(0,20){\circle{14}} \put(0,20){\line(1,1){5}} \put(5,20){\line(0,1){4.5}} \put(0,20){\line(1,0){5}} \put(0,24){r} \put(2,16){x} \put(7,22){y} \end{picture}$
More precisely I am drawing a triangle for any given angle $\theta \text{ (theta)}$ that I want to make. The trig functions take this angle, and tell you the ratio of different values. Notice that in this triangle we have the hypotenuse and two other sides, where the hypotenuse is just the radius. Because the radius is always the same in a circle, the hypotenuse of any triangle I draw with this example circle will be the same. Only the lengths of the other two lines change as the angle changes.

The sine function tells you the ratio of the length of the the side that the angle is facing, y, to the radius r. Cosine tells you the ratio of the length of the side that the angle is adjacent to, x, to the radius r. So math people say:
$\sin{\theta}=\frac{y}{r}$
$\cos{\theta}=\frac{x}{r}$
There are 4 more possible arrangements of these three important numbers, which the other trig functions tell you, but these two are the most important ones to understand because all of the others can be made from these.

Now to combine all of these ideas together we need some simple examples. As math people we like to use $\pi$ to do angles and not degrees. So if we want to ask, what is
$\cos{\pi}$ then we just have to look at the ratio of the x to the r value at the angle $\pi$
$\setlength{\unitlength}{1mm} \begin{picture}(140, 40) \put(0,20){\circle{14}} \put(0,20){\line(-1,0){7}} \put(0.3,20.3){r} \end{picture}$
Well in this case r and x are the same, so $\frac {x}{r}=1$
Next time will be infinite series and sequences.