The transition from understanding angles in radians from degrees is a strange one for many students. Why is it that just after they spend all of this time learning how to measure angles in degrees do they now have to learn an entirely new form of measure?? Much of our folk understanding of angles is measured in degrees - there are 180 degrees in a triangle and straight line, and 360 degrees in a circle. But, as it turns out, we need a new unit of measurement to describe angles if we are to use that measure in some more advanced contexts.
The way we are originally taught to measure angles was a convention made up by the Babylonians since they had a base-60 numbering system. The number 360 was very important to the Babylonians, and so they decided to say that that was the number of degrees in a circle. If 100 was important to them, it would seem like they could have just as easily said that there were 100 degrees in a circle, and all of our subsequent calculations would by based on that standard.
Question: If there are 100 degrees in a circle, how many degrees are there in a triangle? How about a right angle? Did you need to make any assumptions in arriving at your answer?
This raises an interesting point: Do all numbers that define physical quantities have a certain arbitrariness to them? Is there something special about there being 12 inches in a foot, or 60 seconds in a minute? It turns out that most often these numbers are just conventions that were made up by some culture or person and then just continued onwards over time. I recently learned that Tibetans use a measure of one finger width to represent their fundamental unit of measure when drawing images, and that there are 12 such finger widths in a face.
Radians, however, represent a unique example of where there is an objectivity to a standard of measurement which most units don't have. This comes up when we start considering the trigonometric functions like sin(x) and cos(x), and particularly the use of infinite polynomial series to represent functions.
THE UNIT CIRCLE
The definition of a radian has to do with the definition of the circumference of a circle. To understand where a radian comes from, we consider a circle which has a radius of ONE and which is centered at the origin. This circle is called the unit circle, and its equation and graph are:
A radian is defined as the length of the unit circle's circumference that an angle will cut off. So, since the unit circle has a circumference of 2pi, an angle that scopes out the entire circle, which we would normally call 360 degrees, we now call 2pi. This is the actual length of the circle's arc that this angle will cut out if we took a tape measure and wrapped it around the circle. Similarly, if we were to consider an angle that scoped out half of our circle, which we would traditionally call 180 degrees, we now call that angle measure PI, since that is the length of the arc being scoped out.
For most purposes, being able to move back and forth between degrees and radians is as simple as using the conversion factor that 2pi radians equals 360 degrees. So, if I told you to find the number of radians contained in an angle of 40 degrees, we would set up the proportion below and solve for x:
Question 2: Find the number of radians in the following degree measures:
- 30 degrees
- 90 degrees
- 150 degrees
Question 3: Find the number of degrees in the following radian measures:
- pi/2
- 4pi/6
INFINITE SERIES
The reason why we use radians to measure angles has to do with an extremely important yet difficult area of mathematics known as infinite series. In high school mathematics, there is a very special type of function we treat often known as a polynomial. It is basically the addition of a certain number of terms, where each term represents a different power of our variable. For instance, the following are polynomials:
To add to our confusion, we need to also remember that "degree" is a way that we classify different types of polynomials that has NOTHING to do with the word "degree" used to measure angles. A second-degree polynomial, for instance, is a polynomial with an x-squared as its highest power. A tenth-degree polynomial is a polynomial with an x^10 as its highest power.
In general, a polynomial is the addition of a whole bunch of individual powers of the variable, and we can write this in a compact form called sigma notation. In sigma notation, a general polynomial can be expressed as:
where M is the degree of the polynomial. A very important result in mathematics is that many types of functions that are NOT polynomials can be expressed as polynomials of an INFINITE degree, which can be written as:
What this means is that our polynomial keeps adding more terms with higher and higher powers of x, and this can be used to create any function that doesn't have any super sharp corners or breaks in it. This is true of the trigonometric functions sine and cosine, which can be approximated as the following infinite series:
But what value should we use to measure our angle "x"? To get the right answer, we must use radians in the above polynomial approximations. If you were to try to use degrees, your answers would start to get very very large, but we know that sine and cosine can never get bigger than 1 or less than -1. Radians, it turns out, are the objective measure of an angle that we can use in series approximations to get valid results for sines and cosines. In fact, there aren't any situations where we HAVE to use degrees, but here is a situation where we HAVE to use radians, making me feel that it would be best if we never learned degrees in the first place. Who knows anybody from Babylon, anyway?
Question 5: Based on the discussion above, calculate an approximate answer to the following expressions using the first 5 terms of the series approximations for sine and cosine.
