BPuhl’s Blog

A little bit of everything without actually being much of anything

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.

For example:

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).

Well holy cow!  That was sure a lot of work to get to a point which doesn’t actually involve either math, or flying.
What I realized, and try to occasionally remind myself, is that there are times in life when you have a goal, and you can see the straight path to get where you want to be.  And then, “life happens”…  Or as some people may describe, you have to “take an unexpected detour”.  These unexpected detours can seem frustrating, and make you feel like you’re completely “off track”, or “spinning your wheels”, or generally way off course from where you want to be going.
When that happens, I try to stop and remember…  that just because you’re off track…even if you’re off track by what seems like a huge amount (30 degrees is a huge course change!) – It doesn’t necessarily cause a huge change in how far you need to go to achieve your goals (or in our pilots case, how long it takes to get there)
One last random note:  When you’re at the furthest distance “off course”, just before you get to turn back towards your goals…  If this were the plane that took a detour of 30 degrees (the max), how far away from his straight line path would he get (the distance from the peak of the triangle back down to the straight line course)?  86 miles!  When you look at it that way, he’s nearly 90 miles “off course” when he only should have gone 150 miles total.  That’s one heck of a detour, but when he turns back towards his objective, by the time he gets there it only added about 15%… 
Maybe those detours in life aren’t that bad after all?

3 Responses to “Law of Cosines in Life”

  1. It’s not the destination that matters but the journey. :-}

    • BPuhl said

      You apparently were never stuck riding in the middle seat of the back of the station wagon, between your brother and sister, driving through the farmlands of central California, trying to get to Disneyland.

  2. Rose said

    When in the world did you live anywhere else but where your parents live now? You didn’t have to go through central California to get to Disneyland. Or am I missing the point and you were just being relative in that statement? (I’m not really missing the point; just had to give you heck).


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