Why even use degrees at all? Would radians not do the job for all math?

[Lazy Einstein]

With an increasing love for Mathematics, I am trying to teach myself more and more everyday.

I am in an Avionics technician program in college. I have learned all about Trigonometric identities, Pythagorean's theorem, reading Cartesians and complex plane graphs, and everything in elementary Algebra.
Through studying Mathematics and Physics and my own time I have come to learning about "the why of sine, cosine, and tangent". This had me looking at the fact that I can use the universally taught 360 angular degrees of a circle or it's less used equivalent reference of radians.

I get that π = C/d(Circumference over diameter equals pi), θ = s/r(Arc length over radius equals subtended arc in radians), and π = C of a circle with d = 1. So a r = 1 circle has a circumference of 2π.

Radians have a reason for being used as a reference with circles; however, it seems to me that 360 degrees is just an arbitrary reference that is no longer needed.

Am I wrong? Does the use of 360 angular degrees still provide a more elegant reference system for some Math that radians wouldn't? Is it just because radians are a pure number but degrees have a unit association?

Hope my question is clear.

Cheers

 

[daon2]

You are not wrong, degrees, gradients and other measurements are arbitrary. When initially defined in trigonometry radians are also an arbitrary choice when exclaiming the unit circle is "important". It's how many radii have been swept out by a rotation

 

[JeffM]

Radian measure is more convenient for many mathematical purposes, but degrees are easier to work with for any practical purpose where you do not require fractions of degrees to ensure sufficient accuracy.

 

[HallsofIvy]

The main reason for using degrees rather than radians is that an integer number of degrees gives better accuracy than an integer number of radians- so people who prefer working with integers do better with degrees.

 

[Deleted member 4993]

From the Math Forum:

Date: 2 Jan 1995 16:34:20 -0500
From: Dr. Ken
Subject: Re: Origin of degrees

I'm glad you asked this question, because I've been wondering it myself. I figured it had something to do with the Babylonians, who used a base 60 number system. But it sure took a lot of digging in several books to find out anything concrete about it.

I finally found what I was looking for in a book called "A History of Pi" by Petr Beckmann, a mathematician from Czechoslovakia. Here's the passage:

In 1936, a tablet was excavated some 200 miles from Babylon. Here one should make the interjection that the Sumerians were first to make one of man's greatest inventions, namely, writing; through written communication, knowledge could be passed from one person to others, and from one generation to the next and future ones. They impressed their cuneiform (wedge-shaped) script on soft clay tablets with a stylus, and the tablets were then hardened in the sun. The mentioned tablet, whose translation was partially published only in 1950, is devoted to various geometrical figures, and states that the ratio of the perimeter of a regular hexagon to the circumference of the circumscribed circle equals a number which in modern notation is given by 57/60 + 36/(60^2) (the Babylonians used the sexagesimal system, i.e., their base was 60 rather than 10).

The Babylonians knew, of course, that the perimeter of a hexagon is exactly equal to six times the radius of the circumscribed circle, in fact that was evidently the reason why they chose to divide the circle into 360 degrees (and we are still burdened with that figure to this day). The tablet, therefore, gives ... Pi = 25/8 = 3.125.

So that's who gave us the 360 degrees in the circle. See, assignment of degree-measure to angles is somewhat arbitrary. Some choices are more natural than others, though, and when you're working in base 60, 6x60 is a pretty natural choice.

As a sidenote, the actual ratio that the Babylonians talk about is 6r/(2r*Pi) = 3/Pi, which is about 0.95493. They say it's 24/25 = .96. And you might ask why we chose Pi as the letter to represent the number 3.141592..., rather than some other Greek letter like Alpha or Omega. Well, it's Pi as in Perimeter - the letter Pi in Greek is like our letter P.

I hope this helps answer your question. Write back if you have more!

-Doctor Ken, The Math Forum
Check out our web site! http://mathforum.org/dr.math/