Why do we teach students about radians?

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.

CircularSector_1000

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.

green lineSo, 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.

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