In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Cody Jacobs. His topic, from Precalculus: using radians to measure angles instead of degrees

How could you as a teacher create an activity or project that involves your topic?

How can this topic be used in your students’ future courses in mathematics or science?

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

Desmos.com is yet again another great technological resource to use when introducing radians to the classroom. There is a great activity call “What is a Radian?” That introduces an activity student can do using a plate and folding it into different sections. I actually believe this is how I was introduced to radians in high school. The second part of the activity on desmos after you finish with the plate introduction, is asking students key questions. How many radians are in a certain degree measurements? How many degrees are in a certain radian measurement? As always desmos is still great at introducing radians and lets you easily monitor your students progress.

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student A’Lyssa Rodriguez. Her topic, from Precalculus: using radians to measure angles instead of degrees

D1. How did people’s conception of this topic change over time?

Babylonians came up with the degree system. For their number system they liked to use the number 60 and multiples thereof. Therefore they decided on the number 360 and each number represented a degree in a circle. This number was completely arbitrary and was simply a matter of preference by the Babylonians. Although this makes handling circles and angles seem easier, due to it being an arbitrary number, it makes degrees unnatural. So the deeper concepts in math needed a more natural number. Radians are that more natural measurement we needed. Using the length of the radius of any circle and wrapping around the outside of that circle, one can see that it almost completely goes around the entire circle 6 times. To make up for what is left we multiply the radius by 2pi. Thus the equation for the circumference of a circle is C = 2 πr. This is the reason and the change over time for the use of radians instead of degrees.

C2. How has this topic appeared in high culture (art, classical music, theatre, etc.)?
Pure tones are found in music. Regardless of other musical properties such as amplitude or the time relation to other sound waves (phase), these tones will have a consistent sinusoidal sound wave. The sine function used to measure these waves use radians. Although degrees are technically possible, this function is most accurate when using radians. According to Mathematics and Music: Composition, Perception, and Performance by James S. Walker and Gary W. Don, the formula that can be used to determine the oscillation for a tuning fork is y= Asin(θ) where θ is measured in radians and is equal to 2 πvt+ θ0 and θ0 is the initial value of θ when t=0. So y = Asin(2 πvt+ π/2).

B2. How does this topic extend what your students should have learned in previous courses?

In previous courses, students use degrees to measure angles and to refer to circles. Even activities outside of the classroom, such as snowboarding, use degrees. This was easiest and best for learning purposes, up until this point that is. Now that trigonometric functions will be introduced, the circle will be studied more in depth, and more real life situations will be given, it is necessary to use radians instead of degrees. The calculations will become more accurate in some cases, some even easier, and it is essential to use a more natural number. This topic merely adds on to what the students already know about angles but also makes them think about it in a different way. One way their previous knowledge of degrees will be extended is by learning to convert from degrees to radians and back again.

# Angular size

Source: http://www.xkcd.com/1276/

Throughout grades K-10, students are slowly introduced to the concept of angles. They are told that there are $90$ degrees in a right angle, $180$ degrees in a straight angle, and a circle has $60$ degrees. They are introduced to $30-60-90$ and $45-45-90$ right triangles. Fans of snowboarding even know the multiples of $180$ degrees up to $1440$ or even $1620$ degrees.

Then, in Precalculus, we make students get comfortable with $\pi$, $\displaystyle \frac{\pi}{2}$, $\displaystyle \frac{\pi}{3}$, $\displaystyle \frac{\pi}{4}$, $\displaystyle \frac{\pi}{6}$, and multiples thereof.

We tell students that radians and degrees are just two ways of measuring angles, just like inches and centimeters are two ways of measuring the length of a line segment.

Still, students are extremely comfortable with measuring angles in degrees. They can easily visualize an angle of $75^o$, but to visualize an angle of $2$ radians, they inevitably need to convert to degrees first. In his book Surely You’re Joking, Mr. Feynman!, Nobel-Prize laureate Richard P. Feynman described himself as a boy:

I was never any good in sports. I was always terrified if a tennis ball would come over the fence and land near me, because I never could get it over the fence – it usually went about a radian off of where it was supposed to go.

Naturally, students wonder why we make them get comfortable with measuring angles with radians.

The short answer, appropriate for Precalculus students: Certain formulas are a little easier to write with radians as opposed to degrees, which in turn make certain formulas in calculus a lot easier.

The longer answer, which Precalculus students would not appreciate, is that radian measure is needed to make the derivatives of $\sin x$ and $\cos x$ look palatable.

Source: http://mathworld.wolfram.com/CircularSector.html

1. In Precalculus, the length of a circle arc with central angle $\theta$ in a circle with radius $r$ is

$s = r\theta$

Also, the area of a circular sector with central angle $\theta$ in a circle with radius $r$ is

$A = \displaystyle \frac{1}{2} r^2 \theta$

In both of these formulas, the angle $\theta$ must be measured in radians.

Students may complain that it’d be easy to make a formula of $\theta$ is measured in degrees, and they’d be right:

$s = \displaystyle \frac{180 r \theta}{\pi}$ and $A = \displaystyle \frac{180}{\pi} r^2 \theta$

However, getting rid of the $180/\pi$ makes the following computations from calculus a lot easier.

2a. Early in calculus, the limit

$\displaystyle \lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1$

is derived using the Sandwich Theorem (or Pinching Theorem or Squeeze Theorem). I won’t reinvent the wheel by writing out the proof, but it can be found here. The first step of the proof uses the formula for the above formula for the area of a circular sector.

2b. Using the trigonometric identity $\cos 2x = 1 - 2 \sin^2 x$, we replace $x$ by $\theta/2$ to find

$\displaystyle \lim_{\theta \to 0} \frac{1 - \cos \theta}{\theta} = \displaystyle \lim_{\theta \to 0} \frac{2\sin^2 \displaystyle \left( \frac{\theta}{2} \right)}{ \theta}$

$\displaystyle \lim_{\theta \to 0} \frac{1 - \cos \theta}{\theta} = \displaystyle \lim_{\theta \to 0} \sin \left( \frac{\theta}{2} \right) \cdot \frac{\sin \displaystyle \left( \frac{\theta}{2} \right)}{ \displaystyle \frac{\theta}{2}}$

$\displaystyle \lim_{\theta \to 0} \frac{1 - \cos \theta}{\theta} =0 \cdot 1$

$\displaystyle \lim_{\theta \to 0} \frac{1 - \cos \theta}{\theta} =0$

3. Both of the above limits — as well as the formulas for $\sin(\alpha + \beta)$ and $\cos(\alpha + \beta)$ — are needed to prove that $\displaystyle \frac{d}{dx} \sin x = \cos x$ and $\displaystyle \frac{d}{dx} \cos x = -\sin x$. Again, I won’t reinvent the wheel, but the proofs can be found here.

So, to make a long story short, radians are used to make the derivatives $y = \sin x$ and $y = \cos x$ easier to remember. It is logically possible to differentiate these functions using degrees instead of radians — see http://www.math.ubc.ca/~feldman/m100/sinUnits.pdf. However, possible is not the same thing as preferable, as calculus is a whole lot easier without these extra factors of $\pi/180$ floating around.