Overview
For my semester project, I will discuss how triangles and angles helped mathematicians develop the unit circle. This semester I enjoyed finding interactive ways to explain the derivation of the unit circle and how it can be helpful for us today. Understanding its history and creation will help students understand how to use it more effectively in the real world.
The unit circle has been around for centuries. It dates back to the 2nd millennium BC in Egyptian and Babylonian mathematics. One may ask, "Why was the unit circle used way back then?" or "Why is the unit circle so important?" The purpose of this website is to explain the unit circle's history, its geometric origins, and applications. Simply put, the unit circle was created to reflect certain discoverable relationships and mathematical principles. It allows us to easily solve for the sine, cosine, or tangent of some useful angles. The unit circle is far more valuable than we often give it credit for.
One might also ask, why does it have to be a circle with a length 1? Stapel explained the answer to this question well: "To simplify computations, mathematicians like to fit an angle's triangle into a circle with radius r = 1. Because the number 1 is called "the unit" in mathematics, a circle with a radius of length 1 is called 'the unit circle'. Once the hypotenuse has a fixed length of r = 1, then the values of the trig ratios will depend only on x and y, since multiplying or dividing by r = 1 won't change anything. Only the values of x and y will matter" (Stapel). All the trigonometric relationships are the same no matter what radius it is. However, we can see that mathematicians selected the circle with radius 1 because it simplifies the amount of work to solve for x and y values.
The unit circle is a fundamental principle that we learn in trigonometry classes because it helps us understand geometric relationships, and it gives us the tools to solve far more complicated problems. Whether the unit circle is used like a compass for navigators, or for handyman to determine the height of a ladder, it will continue to be useful for all of us for years to come.
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