# Cathy O’Neil on Trig

Cathy O’Neil published this awesome piece about trig on her blog yesterday:

Fuck Trigonometry

It created quite a conversation. Yay!

Her husband’s comment at the end of the post caught my attention and I spent most of yesterday sort of daydreaming about his point:

When I mentioned my hatred of trigonometry to my husband, he countered with an argument that wasn’t mentioned so far. Namely, that we have really no reason to teach high school kids any given thing, so we just choose a bunch of things kind of at random. Moreover, he suggested, if we remove trig, then meeting people at an airport would just elicit some other reason for hating math. We’d be simply replacing trig with some other crappy topic choice.

I think I’m leaning towards agreeing with him. I’m certainly not sure I could make a convincing argument about why trig *needs* to be taught. In fact, with 3D printing and maybe even Zometool sets becoming cheaper and easier to find, my vote would probably be to try more fun geometry projects before diving into trig. Our Gosper curve project, for example, is something that I think kids would find more entertaining that trig identities:

Exploring the Gosper Curve

The passion in the conversation around Cathy’s post also surprised me a little – I didn’t realize that so many people had such strong feelings about trig! Most of the internet math flame wars I see are about addition or fractions – watching people fight about trig was so refreshing 🙂

Thinking back to my high school trig class with Mrs. Kovaric yesterday I honestly couldn’t remember really having any strong feelings one way or another. Without any strong opinions to fight about (ha ha) I started thinking about some fun math ideas related to trig that I’d learned either in high school or in college. Not reasons to teach trig, for sure, but definitely more fun than memorizing identities!

(1) The Extended law of Sines

One idea everyone sees in trig class is the law of sines – in any triangle ABC, A / Sin(A) = B / Sin(B) = C / Sin(C). Pretty neat relationship, but if these three expressions are all equal to each other is their value special? Turns out that it is:

(2) Stewart’s theorem

This is a cool theorem which gives the length of a line segment from a vertex of a triangle to the opposite side. As with the extended law of sines, this theorem is something that I found in Geometry Revisited in high school. The proof (that I know) involves the law of cosines:

Also, the law of cosines came up in a surprising way in an introductory geometry talk I had with my older son this past school year. This conversation was an unexpected (to me) way that you could talk about the ideas behind the law of cosines in geometry class:

When we accidentally derived the law of cosines

(3) The sum of the inverse squares

Using the Taylor series for Sin(x) and the fact that the roots are integer multiples of $\pi$, you can prove that:

$1 + \frac{1}{4} + \frac{1}{9} + \ldots + \frac{1}{n^2} + \ldots = \frac{\pi^2}{6}$

It was incredibly cool to learn that there was a known formula for all of the inverse even powers (solved by Euler in the 1700s, if I remember right), but that a closed form for the odd powers greater than 1 was not known. This is a neat example of an unsolved math problem that high school students can understand and even play around with a little. I’ve always hoped to see a closed form solution for the sum of the inverse cubes.

Another fairly famous trig-related sum problem that blew me away in high school is this incredible sum:

Let $x_n$ be the $n^{th}$ positive solution to the equation $x = Tan(x)$. Find $\sum \frac{1}{(x_{n})^{2}}$

The particularly amazing thing about this problem is that you can find the sum even though you can’t write down a closed form for any of the expressions that are in the sum!

(4) A surprising integral

I went to college planning on majoring in aerospace engineering – that’s what you do with math, right 🙂

Sitting in an introduction to complex analysis class my freshman year, I ran across this interesting little problem:

Seeing this problem made me want to major in math rather than engineering – it was absolutely amazing to me that $\pi$ and $e$ could be connected in such a seemingly mysterious way.

(5) Circles on a sphere

This one is the one and only time that I’ve used trig directly at work (probably more than 10 years ago, though I don’t remember the exact timing).

One of the guys in our office who thinks about hurricane insurance had a list which gave the latitude and longitude of the center of every hurricane that hit North America for the last 50 or so years. The list had coordinates for the center in time increments of 6 hours. The question he wanted to answer was relatively simple: given a specific latitude / longitude (say Miami or New York City, or something) how many Hurricanes had come within a given distance of that city (50 miles, 100 miles, . . . .).

He’d tried to write a really quick and back of the envelope program to answer this question but it was giving answers that seemed really wrong. To calculate distance correctly you need a little bit of trig because you have to factor in how far north you are. Adjusting the distance formula for a given latitude helped him get to the right answer. There were a few other little math-related tricks in the program, too, such as checking whether or not the path between two points came within the desired distance even if the endpoints were outside of the distance. Without trig, the distance calculations in this project were easy to get wrong.

Anyway, not a list of reasons to teach trig, but rather just a few fun trig-related things that Cathy’s post got me thinking about. Hopefully slightly more fun than memorizing identities 🙂

Although if you’ve made it this far and do like trig identities, though, a recent Terry Tao post should be right up your alley:

A “cute” differentiation identity