Random musings: Pi a la mode

In honor of Embrace Your Geekness Day!

This post is based on one of my t-shirts:

I had kind of looked at it, but not really until someone asked me about it when I wore it the other day, so I'll explain it to you.

We'll start with the three right triangles. A right triangle is any triangle in which one of the angles is equal to 90 degrees.The first shows the basic trigonometric definitions of the right triangle. In this case the angle in question, x, is in the lower left hand corner. The three sides are called the opposite, the adjacent and the hypotenuse. The longest side is the hypotenuse (which is also one of the sides next to the angle), the side opposite the angle in question is the opposite side and the other side next to the angle is the adjacent side. The basic identities are sine x equals opposite divided by the hypotenuse or in this example, sine x = opposite divided by 1 or sine x = opposite. Cosine x equals adjacent divided by the hypotenuse or cosine x = adjacent divided by 1 in our example, or cosine x = adjacent. The tangent is defined as the opposite divided by the adjacent or tangent x = sine x/cosine x, abbreviated tan x=sin x/cos x. As you may remember from your math classes, the slope of a line is equal to the rise over the run. In our case, the rise is equal to sin and the run is equal to cos, so the slope of the line is equal to sin/cos, which is also the tangent. So, the tangent is equal to the slope of the line. And the largest triangle helps to make this clear since the rise is the tangent and the run is equal to 1 we get slope = tan x/1 or slope = tangent.

Before we move on to the other two triangles, we'll introduce the three inverse trigonometric functions: the cosecant, the secant and the cotangent. These three functions are simply the inverse of the other three functions: The cosecant is the hypotenuse divided by the opposite or csc x = 1/sin x, the secant is equal to the hypotenuse divided by the adjacent or sec x = 1/cos x. And the cotangent is equal to the adjacent divided by the opposite or cot x = cos x/sin x or cot x = 1/tan x. For those of you who are interested, the approximate vales of the trig functions in our example are: sin x = 0.84147098, cos x = 0.54030231, tan x = 1.55740770, csc x = 1.18839511, sec x = 1.85081570 and cot x = 0.64209262

Now back to the three triangles. They represent the three Pythagorean Trigonometric Identities. For those of you who don't know, or don't remember, the Pythagorean Theorem it goes like this...in a right triangle the sum of the sides squared is equal to the hypotenuse squared or a^2 + b^2 = c^2 (where ^2 means squared), or in the case of our smaller triangle sin^2 x + cos^2 x = 1 which is the first of the three identities. The three identities are:

sin^2 + cos^2 = 1

If you divide both sides by sin^2, you get:

sin^2/sin^2 + cos^2/sin^2 = 1/sin^2 which is:

1 + cot^2 = csc^2 (represented by the middle triangle)

If you divide both sides of the original equation by cos^2, you get:

sin^2/cos^2 + cos^2/cos^2 = 1/cos^2 which is:

tan^2 + 1 = sec^2 (the largest triangle)

Well, that's all fine and dandy but what the heck does all of that have to do with pi? I'm glad you asked! Pi is defined as the circumference of a circle (the part around the edge of a circle...if you run on a circular track you're running around the circumference of the circle) divided by the diameter or pi = C/d and since the diameter is twice the length of the radius pi = C/2r. Now if you multiply both sides by 2r you end up with C = 2 pi r. If you look at the picture and imagine a full circle you can see that the radius (the line from the center of a circle to a point on the circle) is equal to 1 and our equation simplifies to C = 2 pi and since we're looking at half a circle, C/2 = pi, so the length of our curve is equal to pi radians (and the length of the arc within the triangles is 1 radian). I'm not going to get into radians in this article but suffice it to say that the angle x is approximately equal to 57 degrees, 17 minutes and 44.8 seconds.