“If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.” — John Louis von Neumann
It would not be fair of me to throw a formula in your face and dub it amazing without supplying the hard work behind it. (You saw this when we discussed the area of a disk.) Therefore, over the course of the next four blogs, I will do just that. I will not, however, be doing it alone. You, too, will join me on my journey to derive the most amazing formula in all of mathematics, and hopefully, you will acknowledge precisely why it acquired such a name. With that in mind, welcome to the grand finale blogs!
Like any good mathematician, we need a solid mathematical foundation before we can derive anything. This blog and the next three blogs will make sure you understand everything needed to prove this formula. Think of these blogs as less of a math lesson and more of a discovery into the unknowns of your mathematical knowledge. Today, we discuss the very fundamentals of trigonometry, the study of triangles. For those of you who have taken the class and have met it with success, you will not need this supplement. This blog is solely for the benefit of those who may not have had the advantage you had.
To begin, triangles are special geometric objects because they have many properties unique to them. (For example, the sum of the angles inside a triangle must add to 180 degrees.) Some triangles, however, are more special than others. A triangle that contains a ninety degree angle in it, called a right triangle, is the most-specialized case. (See below.)
For definition purposes, we as mathematicians call the side opposite the right angle (the longest side) the hypotenuse. The other two sides are called the legs.
For many years, mathematicians sought to describe relationships between the legs and hypotenuse of a right triangle and the angles formed inside that right triangle. What ensued became known as the trigonometric functions. To be honest, there are dozens of these trigonometric functions, but for the purposes of this blog, we will focus squarely on two of them. One is called the sine function; the other is called the cosine function.
Consider an angle inside a right triangle other than the right angle. (It does not matter which angle you choose.) For standard purposes, let us call this angle θ. Now, consider the length of the side of the triangle opposite to θ (the side of the triangle that does not form the angle θ) as well as the hypotenuse of the right triangle (the side of the triangle opposite the ninety degree angle). The sine function, abbreviated sin, of the angle θ is defined to be equal to the ratio of the length of the opposite side of the triangle to the length of the hypotenuse of the right triangle. Symbolically, sin(θ) = opp/hyp.
To reinforce, the sine function takes a value for some angle (not the right angle) in a right triangle and converts it to a useful ratio of lengths between the opposite side and hypotenuse of that right triangle. For instance, in the diagram above, the sine of the indicated angle could be found by measuring the length of the opposite side and dividing it by the measured length of the hypotenuse.
The cosine function is described in a similar manner. Again, consider an angle θ in a right triangle, where θ is not the right angle. Consider the length of the side adjacent to that angle (the side of the triangle that helps to form θ but is not the hypotenuse) as well as the length of the hypotenuse. The cosine, abbreviated cos, of the angle θ is defined to be equal to the ratio of the length of the adjacent side to the length of the hypotenuse. Symbolically, cos(θ) = adj/hyp.
To reinforce, the cosine function takes a value for some angle in a right triangle (not the right angle) and converts it to a useful ratio of lengths between the adjacent side and hypotenuse of that right triangle. For instance, in the diagram above, the cosine of the indicated angle could be found by measuring the length of the adjacent side and dividing it by the measured length of the hypotenuse.
So to recap what we have learned, just remember that the lengths of the sides of a right triangle can be related to the angles formed inside that right triangle through trigonometric functions. Among these trigonometric functions are sine and cosine. For an angle θ (not the right angle), the sine function is defined as sin(θ) = opp/hyp, and the cosine function is defined as cos(θ) = adj/hyp. For the most part, this summarizes all the trigonometry you need to know in order to complete the derivation of the most amazing formula in all of mathematics. One catch to this summary, however, still remains: Mathematicians do not measure their angles in degrees.
Rather than be sensible, mathematicians prefer to describe their angles in a unit called “radians” (abbreviated rad). I will not talk much about radians on this blog other than the fact that they make our lives so much easier as mathematicians. (If you want to see the ideas behind radians, click here.) The conversion equality between degrees and radians is the following: 180° = π rad.
At any rate, the morale of the story is this: Before plugging in values for angles into sine and cosine, convert those angles into radians!
Below I have included a summary exercise to ensure your comprehension of everything I have taught. It is not meant to be impossible, but rather, it is meant to incorporate everything you have learned in a way that leads to your own exploration of mathematics. If you have no idea where to start, just leave a comment, and I will guide you in the right direction. It is imperative that you have a solid understanding of these trigonometric concepts.
Prove the following trigonometric identity: sin(θ) = cos(π/2 – θ). (Hint: Recall that the sum of the angles inside any triangle must equal 180 degrees, or π radians.)