Trigonometry

By Bobby Neal Winters

Hurt so good

Come on baby, make it hurt so good

Sometimes love don't feel like it should

You make it hurt so good

--John Mellancamp

Before the seventh or eighth grade I would not have said I was good at math.  I was good at science.  I had a great memory. I once destroyed an encyclopedia salesman who came by our rural home when I was ten or twelve years old.  He pulled out his product, opened it to a page that featured a picture of a skeleton, and began is his pitch to my mom.

“This can help your son with is science home work,” he said.  “Every part of the body has a scientific name.  They can’t just call a shoulder blade a shoulder blade.”

“Scapula,” I said.

“What?”

“It’s called a scapula.”

He looked a little thrown off.

“Or a breast bone a breast bone,” he continued.

“Sternum,” I said.

“Or the arm bones,” he proffered.

“Radius, ulna, and humerus.”

He went down and Momma smiled.

But that is just memory.  I loathed mathematics as I knew it.  Arithmetic was my enemy.  Multiplication was hard.  Long division was almost impossible.  By crying, I manipulated Momma into doing it for me.  There was--and is--some block in my head that keeps me from doing it.  Professionals have told me that I have dyslexia, but I’ve never been tested.

The tide in the war between mathematics and me began to turn in the seventh grade when the teachers began to introduce elements of algebra into class. When I took algebra in the ninth grade, I didn’t consider it a chore any more.  In my Sophomore year, I took geometry, and it was as if the scales fell from my eyes.  It was mathematics without arithmetic.  

I thought I was in heaven.

The geometry class had seniors in it, and I cleaned their collective clocks. This made mathematics very important to my self-worth.  I wanted more of it.

My chance came the next year when Algebra II and Trigonometry were both offered. This was a problem because Trigonometry requires the skill set taught in Algebra II, which at that time included the algebra of rational expressions and quadratic equations both of which are needed in Trigonometry.

My teacher, Mr. Sloan, told me that I was doing things out of order but they would let me.  Trigonometry was only offered every other year because I went to a small country high school that simply didn’t have the staff to offer it more frequently.  If I was going to get my trigonometry in before I went to college, I’d just have to do it this way.

For these reasons, Trigonometry became a crucible for my mathematical education.  It was hard because you do need Algebra II in order to do Trigonometry.  There were times when I’d cry while doing my trig homework, but this time Momma couldn’t do it for me because she’d never had it. I had to do it myself.

Two things helped. One of these was that trigonometry has a high content of geometry.  The confidence I’d developed in geometry carried over.  The other was that I was committed to this.  My self-image and ego were on the line.  I did my Algebra II homework first to get it out of the way, and then I did my Trigonometry homework twice.  

This is something that I don’t often share with students but maybe I should.  Doing your homework is good, but doing it twice is better.  It might even be more than twice as good.  You repeat the skill and reinforce it, but you know where it is going and you do it with more confidence.

Writing an assignment the second time is something I’d avoided before then even though it had been suggested to me on multiple occasions. My handwriting is terrible.  I print almost everything and even that is terrible. This is one of the reasons I am suspected of having dyslexia.  So the reason all of my teachers wanted me to recopy was the very reason I wouldn’t.  It was hard.

This time my ego was so tied-up in the subject I finally took the advice.  Aesthetically speaking, the results weren’t good on the second draft, but they were better than the first.

(As an aside, my teachers had always told me to just take my time with my handwriting.  While there is a lot of virtue to that, the subsequent years have proven to me more was needed than just that.  I’ve made new copies of my lecture notes from year-to-year, slowly recopying everything. The results are legible, but barely, and I certainly never have achieved a “good hand.”  There are limits.)

So my course in trigonometry was a struggle for me.  It was an example of what is called productive pain. Okay, what is it and what’s it good for?

Trigonometry is the study of triangles.  Triangles are geometric objects, but in trigonometry we use numbers and algebra to study them.  There are two major aspect to the course: practical and theoretical.  

The practical part consists of learning various techniques, including  the Law of Sines and the Law of Cosines, in order to measure the sides and the angles of a triangle from known information.  There are certain situations where you can get back a whole lot more information than you put in.  Students, especially those who are of a practical turn of mind, seem to appreciate this part of the course as it can be immediately applied.

They are not so sanguine about the theoretical part of the course.  Those who’ve had the course will know that I am referring to the various identities one is force to learn, manipulate, and prove to be true.  Students don’t like trigonometric identities.  Indeed, hate is not too strong a word to use here.

The proofs that we make students perform in these identities are far from intuitive.  They are like mazes in that students can make a wrong turn and have a hard time recovering from their mistakes.  Why, oh, why do we subject students to such pain, other than the native sadism?

Well, in my opinion, our native sadism is reason enough because this sort of pain is good for you, but beyond that, these identities are, in the long-run far more useful than the mensuration formulas we teach.  First, we have to use these identities to prove the mensuration formulas.  There is no royal road to geometry, Mister, if Alexander the Great had to learn it, then you do too.

But more than that, these formulas will be seen again in calculus.  They make certain otherwise impossible problems easy.  In addition,electrical engineers will probably take a course in Theory of Functions of a Complex Variable, and these trigonometric formulas pop up again there.

Indeed, I encountered formulas of trigonometry as deeply in mathematics as algebraic topology, and that is pretty deep indeed.

But the productive pain aspect of it was by far the most important part for me.  School, research, and life itself are places where being able to endure this sort of pain are vital.  In Trigonometry, I learned how to do that and picked up some cool formulas while I was at it.