Recent posts about the RAD and DEG modes of HP calculators started me musing about angle units.

Somewhere in my mathematical education (such as it was), I acquired the notion that the radian is a more "natural" or more "mathematical" unit for an angle than is the degree. Certainly the radian has elegance and simplicity, being an arc length equal to the radius of the circle divided by the radius itself. But the degree is almost as simple. Suppose we compare an arc length equal to the circumference of the circle to the circumference itself, and call the resulting unit a "circumian". This may sound moronic, dividing a circle by itself and calling it a unit. But from such a dimensionless unit, subunits can be cut which are rationally related to the whole circle, which the radian can never be. It is, after all, whole circles or rational parts thereof that often are required. Since the "circumian" is too large a unit to measure parts of the circle (the radian is also a little clumsy in this respect), it needs to be broken into smaller parts. The obvious choice might be 100, but someone chose 360 instead. Where did the 360 degrees in a circle come from? Probably from Babylonian (yes, it started in Iraq) and Mayan astronomers, who liked the number's easy divisibility, combined with its approximate correspondence to the number of days in a year. The degree likely derives from the observation that the sun moves against the stars (as seen at sunrise or sunset) about that angle per day around the zodiacal circle of the year. In that sense, for those of us on earth, it is a rather "natural" unit.

The degree, like the foot or pound or hour, is such a familiar everyday unit that it is often thought to have dimension, but actually it is just as dimensionless as the radian. The "circumian" is certainly dimensionless in the same obvious way as the radian, and the unit created by dividing it by 360 is therefore also dimensionless. In normal usage, all units for angles are dimensionless, and the ordinary trigonometric functions require dimensionless arguments.

So where in mathematics was I made to feel that the radian is, in some fundamental way, purer or better than a degree? The answer comes from the calculus, in a way that I never fully appreciated, or have long forgotten. The derivative of the sine is the cosine, right? Well, it depends on the unit of the angle. It is so if angles are expressed in radians, but not otherwise. If the angles are in degrees, the derivative of the sine is not the cosine, but the cosine multiplied by (pi/180). That is not nice! And things get worse. The derivative of the cosine is a negative sine muliplied by (pi/180), and the second derivative of the sine is a negative sine multiplied by (pi/180)^2. The sequence of the derivatives of the sine and cosine, instead of forming the remarkable cycle we are accustomed to, forms a spiral of ever-diminishing values. Bizaare! The number (pi/180), which is of course the number of radians in a degree, starts to pop up in its various powers all over the place in the calculus of trigonometric functions if we try to use degrees instead of radians. Why is this so?

It all has to do with an important limit, the limit of the ratio of the sine (or tangent) to its angle, as the angle becomes very small. This limit is unity -- but only if the angles are expressed in radians. That is, sin(x)/x goes to 1 as x goes to 0 only if x is in radians. If we use degrees, the limit is -- you guessed it -- (pi/180). The radian is the only measure of angle that will give us unity for this limit instead of some other constant. (Try this on your calculator: calculate sin(x)/x for small x, first in degree mode, and then again in radian mode.)

This is a little like the case for the natural logarithms compared to any other logarithms such as the common logs. Just as the number e provides the only base such that e^x divided by its own derivative is unity rather than some other constant, so the radian provides the only angle unit for which that important limit of sin(x)/x is unity rather than some other constant.

Because the limit of sin(x)/x enters into the derivation of the derivatives of the trigonometric functions, the nice clean relationships that we remember depend upon that limit being unity, which requires that the angles be in radians. Thus too the simple polynomial for calculating a sine or cosine from a Taylor series is valid only if the angles are in radians. If degrees are used, the Taylor series for the sine is:

sin(x) = (pi/180)x - [(pi/180)^3]x^3/3! + [(pi/180)^5]x^5/5! .... ,

which reverts to the familiar form if we just convert degrees to radians by letting x (or x') equal (pi/180)x.

So while the degree and radian are both useful in plane and spherical trigonometry, in calculus the radian emerges with an elegance (or radiance!) that the degree can hardly match. Nevertheless, I am glad that my compass, protractor, and sextant are scribed with degrees rather than radians. Since there are 2pi radians in a circle, and pi is irrational, no rational unit based on the radian could be scribed in such a way as to cover exactly the circle of my compass, the half-circle of my protractor, or the 90-degree arc from my horizon to my zenith. I guess we will keep both the degree and the radian, and be glad that our HP calculators have both modes available for most calculations.

The angle as I see it (probably obtusely!),

Cheers, Tom