The importance of the unit circle in mathematics is difficult to overstate. The principles contained within the circle are absolutely essential for an understanding of triangular relationships (trigonometry), which, in turn, allow an understanding of various forms of oscillating patterns (from a mathematical perspective), which means that the unit circle has become essential especially within physics, describing everything from the quantum mechanical wave-properties of an atom to the rhythmic bouncing of a spring.
What could be so very important about what seems on the surface to be a simple circle?
Points on the Circle
The key feature of the unit circle are the points which may be inscribed along its circumference. Because the circumference of a circle is well known to have a value relative to the radius (which in the case of the unit circle, of course, is merely 1) of 2pr, because on the unit circle r=1, it is easy to see that the circumference of this circle is very simple: 2p.
As a result of this knowledge, many other points may be inscribed upon the circle's surface, such as p itself (at exactly 180 degrees around the circle, at point (-1,0)), as well as p/2 (at point (0,1)) and 3p/2 (at point (0,-1). Any points in between these can also be determined based on their relationship with p.
Modulation of Points
Why does this help? Here is where trigonometry comes in.
Having knowledge of a point on the unit circle, it is relatively simple to determine the dimensions of a right triangle inscribed within the circle with its hypotenuse reaching that line and its adjacent side along the x-axis. Then, using simple trigonometry (sine, cosine and tangent), it is easy to come up with every possible value for a single point.
How does one point have multiple values?
Because it is a circle, which is essentially modular by definition. This is the value of the unit circle in quantifying harmonic motion. For any given trigonometric value on a circle, there are an infinite number of possible points on the circle. For example, if asked to find the point which corresponds to sin 1/2, the simplest solution would be p/4. However, there are many other solutions, because one could also merely follow the path around the circle again until reaching that same point once more, only this time with 360 degrees of rotation added to it, bringing the value up to 2p/4. 4p/4 and 6p/4, then, are also correct answers.
Graphing the Unit Circle
The importance of these infinite values does not become truly effective until one attempts to graph them. For any given trigonometric value, graphs show very simple, endless harmonic motion - for sine and cosine functions one finds a repetitively oscillating wave, while for tangent functions there is an endlessly regular sweeping of verticle asymptotes, for every p moving to the left or right.
It is these curves, and the various transformations of them (i.e., f(x) = 3sin(4x) - 4tan) which hold value in both calculus and in the physical sciences, and it is mastery of this concept which provides endless mathematical value, for there are a great many things in this universe which operate according to this very behavior!
For more detailed information, visit Paul’s Calculus Notes.
Isaac M. McPhee - Isaac McPhee was born as a human child in Mt. Vernon, WA, c. 1982; he currently resides in the bustling heart of New York City where he ...
