When are we ever going to use trigonometry?

In the summer of 1994 I worked at an outdoor education center in Montana. We lived in crumbling Forest Service barracks on the north fork of the Flathead River. The Douglas Fir forest swarmed with deer, at night we heard mountain lions. Every week a bus would ramble up the dirt road and empty a rabble of fifth graders versed enough to recognize a bull trout but still thrilled to see otters riffling the river, and recoiling in nervous delight from binoculars when they spotted a grizzly on the adjacent huckleberry mountain. Teaching kids about the outdoors in a place so wild was like teaching art history in Le Louvre.

Because the baracks were in such a state of disrepair, I one night saw the Aurora Borealis. The toilets in the teacher cabin had not quite flushed in some days and we were waiting for a backhoe to dig out and replace the pipes to the brownfield. I woke and wandered down the hall, surprised to find it rather well-lit. Outside, the mist of my sudden exhalation lifted to mix with the waves of light. It seemed much closer than I had imagined. It was indeed just like a postcard, in your hand, right there at hand.

The next morning came the backhoe. The map showed lines here and there but they dug several holes and couldn't hit the pipe. It occurred to me that if I wriggled into the crawlspace under the barrack I might be able to determine the angle at which the pipe began its path and with some trigonometry establish precisely where it exited the building. Sure enough, after brushing aside the cobwebs and a bit of scratching, I exposed the section of pipe heading out, and roughly measured the angle with a navigational compass gleaming in the light of my headlamp. A couple calculations further and we had the line marked with string in the sunlight, and soon the pipe was reached and plumbing restored.

I do sometimes wonder how many times I would have seen the Aurora Borealis had I not used trigonometry to solve that problem. But at least I have a great answer when a student asks what trigonometry is good for. I try not to teach a procedure unless I've given students a reason to use it. There are plenty of interesting problems to solve in the world, and we should by all means give students the most compelling problems we can imagine. While this particular story has not become an investigation for my students to work on in class, there are plenty others, real and invented, which I've created to draw students to light and pattern.