When will I ever need to use trigonometry? (Or, how to avoid avalanche terrain.)

21 August 2009

I sometimes tutor high school math, and as anyone who has done this knows, a common question is ‘But when will I ever need to use blank?’. This came up last year, where the blank was trigonometry.

When will I ever need to use trigonometry? It’s a legitimate question. As an instructor, you could explain how it’s a foundation of mathematics, or how you’ll need to understand it for any career in science or math, or so on… (They’re both true, and are probably close to what I said, but neither are an answer that your student is likely to care much about.) Think about it: what practical problem, in day-to-day life, have you solved by applying trigonometry? (Comment if you have a good one!)

So, when I was backcountry skiing last winter and my friend Erik and I actually applied trigonometry to decide if it was safe to ski a certain slope, we were proud.

It’s generally accepted that if a slope is 30 degrees or less, it’s probably not steep enough to avalanche. Slopes around 40-55 degrees are the most dangerous. We were about to cross a slope, but we weren’t really prepared to be in avalanche terrain: we didn’t have our beacons, probes, and shovels. In short, we wanted to avoid avalanche terrain by staying off slopes that were 30 degrees or greater.

We didn’t have a slope meter, but we did know trigonometry. If we could make a right triangle where one of the legs (i.e., sides not opposite the right angle) was double the length of the other leg, the slope of the opposite side (the hypotenuse) would be less than 30 degrees. (We did need a calculator to figure out exactly what that angle would be.) What could we use to make the two legs? Ski poles!

Figure 1: Identifying a 26 degree slope with two ski poles.

Figure 1: Identifying a 26 degree slope with two ski poles.

Here’s how it works. Plant your first pole vertically, and try to make a 90 degree angle with the second pole beginning halfway down the vertical pole. If the tip of the horizontal pole touches when the poles are at 90 degrees with one another, you’re on a roughly 26 degree slope (Figure 1). If you can’t make a 90 degree angle because the horizontal pole hits the slope too high, the slope is steeper than 26 degrees (Figure 2). If you can make an angle smaller than 90 degrees before the tip of the horizontal pole hits the snow, the slope is less than 26 degrees (Figure 3).

Figure 2: Identifying a slope greater than 26 degrees with two ski poles.

Figure 2: Identifying a slope greater than 26 degrees with two ski poles.

The rule-of-thumb is this: If the poles can form a right triangle with the horizontal pole exactly halfway down the vertical pole, and with the slope as the hypotenuse, the slope is less than 30 degrees and you are probably safe. This is the situation depicted in Figures 1 and 3. If you can’t form the 90 degree angle because the slope is too steep, as in Figure 2, you’re in possible avalanche terrain so make sure you know what you’re doing!

Figure 3: Identifying a slope less than 26 degrees with two ski poles.

Figure 3: Identifying a slope less than 26 degrees with two ski poles.

Finally, you can also use this trick to identify a 45 degree slope (which is absolutely in avalanche terrain). To do that, this time place the horizontal pole at the top of the vertical pole rather than halfway down. If the tip of the horizontal pole just touches the slope when you have a 90 degree angle between the poles, the slope is just about 45 degrees (Figure 4). If it doesn’t touch, it’s less than 45 degrees. If you can’t make a 90 degree angle because the second pole hits the slope too high, you’re on some steep stuff!

Figure 4: Identifying a 45 degree slope with two ski poles.

Figure 4: Identifying a 45 degree slope with two ski poles.

There you have it: a field application of trigonometry. We’re not the first ones to figure this out — I’ve since heard that the Norwegians have been doing this forever. It’s a neat trick.

And of course, there is a lot more to avalanche safety than this. This is a basic rule, but by no means definitive. Be scared of avalanches, and don’t assume that if a slope is less than 30 degrees it can’t slide. If you’re planning to travel in avalanche country take a class, check the avalanche reports, and if you’re not scared of avalanches, watch some videos.

Skiing out from Ken's Cabin
Skiing out from Ken’s Cabin

More photos from this trip.