Should Measurement of Angles Using Degree (and perhaps Common Logarithm as well) be Avoided in Pre-Calculus?

Question

People use degrees and radians to measure angles and though degree measurement is acceptable and is widely used in everyday life, it is not in the International System of Units and mathematically it is much less convenient to use than radians (as an example, think about the derivative of the sine function if the variable is measured in degrees). Since the main purpose for the course Pre-Calculus is to get the student prepared for Calculus, in which the unit "degree" is seldom used, should the measurement in degrees be avoided starting Pre-Calculus, if not at an earlier course, in the first place? In my mind the measurement in degrees is only good for elementary mathematics.

Same argument applies to the Common Logarithm (Logarithm using $10$ as the base). How useful is the Common Logarithm compared to the Natural Logarithm in calculus, or in mathematics in general?

Answer 1

No. You lose a lot of the intrinsic value of the course. People need to be able to deal with degrees in physics class (even one not using calculus) to deal with forces and velocities and artillery and such. This includes both students who go on to calculus AND students who don't. Your question ignores the value of the course for many students who never take calc, but even the calc taking ones need facility dealing with angles in multiple units.

Finally, there is nothing to be scared about dealing with different units and it is one of the most practical important things people will need to deal with in daily life in any technical feild (to include medicine!) Psia, psig, atmospheres, Torr, inches of mercury, inches of water, mm of Hg, Pascals, etc. effing etc! Don't avoid this difficulty, learn to handle it.

[Chemistry class is great for this because even with most of the work being SI, you are forced to do conversions because of different molar masses. But even in chem that is mostly "metric", there is still an expectation to handle different gas laws. And of course in engineering, it is unavoidable to do unit conversions.]

Finally, students are familiar with degrees for angles from pre math class awareness. Forcing them to learn trig and such only in radians means learning two new things at once. This is bad pedagogy.

Answer 2

I think it's good to realize and accept that units of measure are (generally) arbitrary.

Radians are usually introduced via the length $s$ of the arc of a circular sector. This arc length will be proportional to both the radius $r$ and the angle $\theta$, so that $s = k \cdot r \theta$, for some constant $k$. We choose radians to make this constant $1$, but I couldn't really argue with anyone who thinks choosing $k = 1$ is still arbitrary (although I do feel it's the least arbitrary).

While formulas and calculations generally look nicer using radians, everything could still be made to work whether we choose to divide a full rotation into $360$ units, $2\pi$ units, or any other number of units. So really, measuring angles is useful, no matter what you choose as your yardstick.

But probably more valuable than appreciating a "nice" unit of measure, is the ability to transition between units of measure. Because we'll always have different ways of measuring the same thing, there will always be a need to reconcile these superficial differences.

I think this is an important idea: As long as you are talking about the same thing, there's always a way to transition from one viewpoint to another, and express the same idea in equivalent ways.

Granted, I don't spend very much time in any precalc class on changing between different bases of a logarithm, or converting between degrees and radians, primarily because there's not a lot to say about either.