Is the radian a more "natural" unit than the degree?


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


Thanks for the interesting thoughts, Tom. It's things like this that make mathematics interesting to me.


Yes, you could say radians are more 'natural'.

In fact, radians are a "basic unit": they need no conversion to another system before being used as input to trig algorithms calculating sin, cos, tan etc. And the arc (inverse trig) functions naturally deliver a radian value. Employing other than radians in these algorithms would either require preamble or epilogue conversions or have a more complex derivation with conversion constants folded into terms in the infinite series...

Radian use is much like use of Base 'e' for natural logs.

Bill W

San Jose CA USA


There are several misconceptions at work here. It's unfortunate that there is never time in a school math class to delve into these things.

It's commonly stated that the degree is based on the Bablonian calender of 360 days plus a five day holiday, but kiddies, it just ain't so. The actual derivation is from the very early devlopement of trigonometry and involves a convenient number to simplify calculations. Much like we buy eggs by the dozen because 12 is divisable by 6, 4, 3, 2, & 1 without resorting to paper and pencil. An important point in a country market where few of the customers could read, write, or 'rithmetic. I have some articles on this derivation, but would have to refresh my memory to go into detail.

The grad of course, was an attempt at metrification.

Now on to the radian, or "natural" unit.

First, by definiton, an angle is equal to twice the area it sweeps in its unit figure of generation. (remember this definition, not all angles behave in the same manner! More on this later.) In the case of a circular angle (the common garden variety)a bit of math will quickly show that an angle of one radian sweeps an area of one half in the unit circle. The unit circle has an area of pi so a complete circle has 2pi radians.

Now we will get messy. In the case of a hyperbolic angle, the figure of generation is the unit parabola. Here we find that as the angle approaches 45 degrees as measured with a protractor its actual value approaches infinity. A hyperbolic angle is NOT the same animal as a circular angle.

It is also commonly stated that "natural" logs have e as a base. This is not true. Natural logs have no base what so ever. They are defined as an integral, not as the power to raise some base to a certain number.

Ordinary logs may be to any base you like, including complex numbers. We can choose e as a base. Surprise! Log to base e of X equals ln X! How ever we should preserve the distinction that natural logs have no base.

Another common error is the habit of calling natural logs or log base e "Napierian" logs. Actually Naperian logs do not look at all like common logs or ordinary logs. The mthod of calculating them and of using them in calculations does not look at all like ordinary log calculations.

I wrote a program for my HP-48SX to calculate Napierian logs or Nog X and the antilogs.

If any body here has too much time on their hands and desires to be bored stiff I can explain the Nog function later.


Hello Unspellable,

very interesting me this function! (And I have a HP48SX too! :) )



i discovered that, for a calculator, you cant just convert into radians as the preamble for trig functions.

for example, if your calculator has modes, say deg or rad. then in deg mode the sequence 36e20 sin should be 0 (which it is on some hps). however, if you go into radians and calculate 36e20 pi * 180 / sin, then you dont get zero.

so you cant internally just secretly convert to radians and get away with it.

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