The two most commonly used measures for angles are degrees and radians. There are 360 degrees in a full circle (a right angle is 90 degrees), and
Students typically learn about degrees before they learn about radians, which brings up the question: Why learn about radians if degrees are good enough for measuring angles? There are two reasons, but both are grounded in the fact that the radian is a unitless measure, which I’ll explain in the following paragraph.

The radian is defined to be (see the diagram) the ratio of the length of the arc of a circle (indicated by
When
Unless there is a good reason, one is free to use any measure whatsoever for angles, such as degrees, radians, or one of the less common ones. There are two good reasons to use radians:
1. If you are working with the derivative of a trigonometric function, then it is preferable to use radian measure for angles, because then derivative formulas (and limit formulas) are easier. For example, using radians, the derivative formulas for sine and cosine are:
and
However, if degrees are used, the derivative formulas for sine and cosine are:
and
The latter pair of formulas are a pain because of the additional factors, so why wouldn’t we use the simpler first pair of formulas? It’s the laziness principle that is used so often in mathematics … we always do the easiest thing. Therefore, we use radians whenever we are dealing with derivatives of trigonometric functions. (The derivatives of the other trigonometric functions are similarly simplified if radians are used … work them out, if you wish!)
2. In describing rotational motion, one rearranges the definition of radian measure to relate the linear displacement to the angular displacement:
Differentiating both sides of this relation with respect to time (and assuming a constant radius), one obtains a relation between the linear velocity
If
Differentiating the relation from the previous display with respect to time once more, we get a relation between the linear acceleration and angular acceleration for an object moving in a circle:
A similar comment about units is also relevant here: Measuring
Finally, there is something arbitrary about degree measure (why 360? why not 100, or 1000, or some other round number?) that makes it aesthetically not pleasing. On the other hand, there is something aesthetically pleasing about the definition of radian measure, as it is a simple ratio. This makes it a natural measure, wouldn’t you say?
Which leads me to think about the use of the word “natural” in mathematics … it would be nice to discuss this another time.
p.s.: By the way, did you notice that the factors in the formulas for the derivatives of sine and cosine in degree measure are the same as the factor that is used to convert degrees to radians? This makes sense if you think about the conversion from degrees to radians as a scale change (i.e., a change of variable) and then use the chain rule for differentiation.
Update, 20 September 2012: Ted Burke notes in a comment to this post that Euler’s formula
is valid only when the angle
This is another situation in which radian measure is more convenient than degree measure.
Update, 28 February 2016: Parker Harris contributed a very interesting comment, which leads me to doubt that “unitless” is a good term for the radian. Perhaps “ignorable unit” is a better term, as discussed in my response to Parker’s comment.
Thanks very much to Parker for his thought-provoking comment!
(This post first appeared at my other (now deleted) blog, and was transferred to this blog on 22 January 2021.)