Law of Cosines in Life
Posted by BPuhl on March 3, 2009
I’m a pilot. I’m fascinated by airplanes, helicopters, gliders, blimps, and anything else that flies. When I’m not actually flying (which is too often), then I’m reading books or magazines about it. It’s fun, and it’s my distraction from everything else. In fact, I should be working on something else at this very moment, but flying is more interesting…and blogging is more interesting…and it’s 3am anyway, so what the heck right?
I remember reading an article in a magazine a few years ago, that I’ll credit it to Barry Schiff in AOPA Pilot magazine, though I’m not 100% sure that’s accurate. The article was about the law of cosines (oh yeah, did I mention that I like math almost as much as flying?), and how when it comes to planning a flight, the best distance between 2 points may not be a straight line.
Let’s take someone who wants to fly from point A to point B. Pilots know that it’s generally safer to have someplace to land at all times during the flight (just in case). So it may be “better” to fly straight, how much would it cost to take a minor detour in your course to fly near an alternate airport? Graphically, it would look something like this:
The question he posed is, just how inefficient is it to take a detour? Even without doing any math, it’s pretty easy draw a couple of things from the picture:
1) If the angle that you deviate from the straight line course is little, then the distances shouldn’t be much
2) If the angle that you deviate from the straight line course is large, then the total distance you fly will be larger
(everybody say “duh” now) 🙂
Just for examples though, let’s look at some real numbers. Let’s take this typical small plane flight distance of 300 miles at an average speed of 120mph. And let’s figure out just how much further you’d have to go, and how long it would take, if you flew out at 10, 15, 20, and 30 degrees off course. We’ll also do the baseline, of 0 degrees, or going straight from A to B.
Angle From Straight Total Distance (miles) Total Time (min) % Increase 0 300 150 0% 10 305 152 2% 15 311 155 4% 20 319 160 6% 30 346 173 15%Huh… not nearly as big as what you might have thought?
For those that are really curious, remember that Cosine is the adjacent side (in this case 150 miles), divided by hypotenuse (which we want to find). Since we’re simplifying things by having the two halves be equal, we can just use: 300 / Cos(a) to get the total distance flown. Take the total distance flown, divided by 120 mph, to get the total hours (times 60 for minutes).