A math problem and a general life lesson

A former student highlighted a problem that I found too intriguing to resist. It was challenging, but I managed to solve it.

Let me be clear - the point of this post isn't to shore up my ego by demonstrating my ability to do algebra better than someone else. That is not the point at all. I think the student showed just the right judgment in struggling with it for a couple hours, then giving up and searching for an answer. That's not intellectually dishonest -- it's quite smart. I probably would've given up and not ask anyone -- which I did do, to a distressing level, in college.

The point is to show how something intimidating and new can be connected with prior knowledge. College, especially in technical disciplines, is largely about extending this prior knowledge in ways that are new, but connected with what you already know. Don't worry, you'll get practice, training, and experience, which will refine your judgment and improve your ability to solve problems of increasing complexity.

Here's the question:

Here's my handwritten solution. Sorry if the photo isn't great - my scanner is broken.

It took about an hour. Most of that time was spent on wrongheaded paths. Once I realized what would work, it took about ten minutes. By the way, that's probably an optimistic estimate of the amount of time solving a problem/knowing what to do versus the amount of time wandering like blind moles in research, in my own terrible experience.

 Some of my friends might find that ridiculously slow. I suppose I could say I'm rusty, but, to be honest, I never really put as much energy and thoughtfulness into mathematical derivation in school as, say, understanding psychology or international relations. It just never interested me -- or it frightened me.

Remember, the point of this post is to illustrate a point about math, and life. Sometimes, when confronted by something that seems impossible, it can help to look at hints for how to derive something.

Repeating, in case it's not clear from the scan.

In this case, there were two key insights:

1. The solution form looks like solving for tan (x/2) was the penultimate step. This suggested a substitution of a dummy variable equal to x/2.

2. The radical on one side of the equation in the penultimate step suggested a quadratic solution, with tan(x/2) as "x" in the quadratic form of the equation. From that, I was able to figure out what A, B, and C are. (Note that it doesn't matter whether A is positive or negative, but the relative sign of A, B and C are important.)

Based on that, I was able, after some time, to figure out the right approach and the right substitutions to make, and solved it quasi-formally.

Note:
In this case, looking at half-angle and double-angle formulas was briefly counterproductive. I saw a radical in the half-angle formulas, and a radical in the solution, so I thought that was the proper approach. Big mistake - I ended up wasting time just because of a superficial connection.

As with many things, looking for and identifying superficial connections between things can lead us astray.

For this problem, I needed to know how to recognize the solution of a quadratic equation, how to use a quadratic equation to solve for a function (versus a variable like x), how to substitute a variable for another one to take advantage of double-angle formulas, and a sense of which double-angle formulas I should use. (sin 2x is straightforward, while there are three choices for cos 2x; even though they are equivalent, not all are equally useful in a given situation. The handwritten work shows I momentarily chose the wrong one.)

So stuff from Algebra 2 could be put together to solve an algebraically intensive problem from first-year college physics.

Now I think this is generally true, even at the cutting edge of science, and perhaps other human endeavors. We build off of our prior knowledge. We extend our knowledge and tools to dazzling heights -- but, barring something really weird and advanced beyond my ken, every tool and every solution builds by analogy or by logical extension from previous ones.

So the next time you see something like this on a test -- don't panic! Take a deep breath. See if the form of the solution looks familiar. It might pay to make a list of the standard tricks -- in this particular case, substituting to take advantage of double-angle formulas, and solving for a function instead of a variable using the quadratic equation, with particular attention to coefficients.

Don't be afraid to work backwards -- sometimes it makes a lot more sense to do that. For the formal proof, you need to start from the beginning, but there's no rule saying you can't understand a derivation by reverse engineering it. Just practice, and try to see how anything that is "new" relates to things you know already. If you don't know those other, more fundamental things, make sure you do before you tackle the complicated things.

(Aside: I wish I thought this clearly 11 years ago. I would've probably done a bit better in college and gotten more out of it.)

Finally, to my former students -- sorry if sometimes it seemed like you were being thrown into the deep end of the pool when I was teaching. It was tough-- there were students at all different levels, and I think as an amateur teacher I didn't do as good a job as connecting the dots as I should've. Consider this a down payment on that deficit.

Now, left as an exercise to the reader: extending this analysis beyond math into daily life.