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

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?

• But I think that in the US the main goal of Calculus is to get students ready for science (not just mathematics) and the $\log_{10}$ and degrees are used quite a bit in those disciplines. – ncr Feb 7 '18 at 19:14
• @ncr Thank you for the comment! I know very little about other disciplines; my impression is that radiant is more commonly used than degrees in pure mathematics (just as natural logarithm being more common than common logarithm). – Zuriel Feb 7 '18 at 20:56

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.

• Thanks for the answer! As far as science is concerned, my impression is that the SI base unit "kelvin" is less common than "Celsius". Now the question is, when should one use units from SI? – Zuriel Feb 7 '18 at 20:53
• Use both. That would be implicit from my comment so far. – guest Feb 7 '18 at 21:47
• And while we're speaking of chemistry, bond angles are pretty much always measured in degrees, whether in high school courses, or in the introductory survey and organic chemistry courses, or in the more advanced analytical and organic and physical chemistry courses. – Dave L Renfro Feb 8 '18 at 11:42
• @DaveLRenfro Thanks for sharing! I talked to a scientist and was told that in his research, the temperature is always measured in Celsius (though he is an American). Now the question is, since there is a SI base unit (kelvin) for the temperature, why don't even scientists use them? – Zuriel Feb 8 '18 at 13:27
• Scientists (and engineers and technicians and doctors and anyone in real world) uses whatever unit they need to. They are not scared of having different units at different times. Celcius is fine if you are dealing with delta T for thermo, are doing a distillation in the lab, etc. If you need to do a gas law calc, you go to Kelvin. Heck US engineers still need Rankine at times: en.wikipedia.org/wiki/Rankine_scale. Why do you think people can't translate back and forth? – guest Feb 8 '18 at 15:11

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.

• Thanks for your comment! My thought was, since there is an international standard, it should be respected. People may use other units in daily life, but in academics, such as in a maths or science class, units from the international standard should be encouraged, if not exclusively used. – Zuriel Feb 8 '18 at 13:33