Let’s pick this up right where the table left off — Page 189, Lesson 170

Let’s pick this up right where the table left off. I’ve just shown you that if you expand the adjacent side of that triangle to 60 metres, you multiply every opposite side by 60 as well. So I added a third row to the table: for angles z of 1, 2, 5, 10, 15 and 20 degrees, the tangent values are .017, .035, .087, .176, .268 and .364. Multiply each of those by 60 and you get 1.02, 2.1, 5.22, 10.56, 16.08 and 21.84. Now here’s the point I want you to see. Suddenly there’s a very close correlation between the angle z and 60 tan z. A one degree track difference gives you 1.02 nautical miles off track in 60 nautical miles along track. A ten degree track difference gives you 10.56 nautical miles off track in 60 nautical miles along track. Even at 15 and 20 degrees, the relationship stays pretty close. But I want to be honest with you — you should not really need to use the 1 in 60 rule for angles much above 10 degrees. You should never get that far off track in the first place. So why isn’t the correlation perfect as the angles get bigger? There are two reasons. The first is that we intentionally introduced a 5% error to make the arithmetic easier. We use a 1 in 60 rule in the air, not a 1 in 57.3 rule. If you knock 5% off all the figures in the “60 tan z” row, the correlation becomes very close indeed. The second reason is that the perfect correlation comes from using the arc of a circle. The explanation I just gave you considers a right-angled triangle, and the tangent of an angle is not a linear relationship. But up to about 20 degrees, it is very close to linear. Now, as I said before, you don’t actually need to know why the 1 in 60 rule works, as long as you accept that it does and you can apply it in the air. So let’s move on to applying it. Here’s the practical problem. In the diagram, the adjacent side represents 60 nautical miles of distance gone along track. The pilot fixes himself 8 miles left of track. He therefore has a track angle error of 8 degrees left. But fixes do not always come conveniently at 60 mile intervals. In any event, 60 nautical miles at Warrior speeds corresponds to 30 to 40 minutes, depending on ground speed, and that is too long an interval between fixes. So we need to be able to use the 1 in 60 rule when the along-track distance is something other than 60. The problem is a simple one — it’s a matter of similar triangles. Suppose we have only gone 30 miles along-track. For the same track angle error of 8 degrees left, we will only be 4 miles left of track. Four miles in 30 is obviously the same as 8 in 60, so the angle is still 8 degrees. Ten miles in 120 is obviously the same as 5 in 60, so the angle is 5 degrees, and so on. So the method is this: we simply take the ratio of 60 nautical miles to our along-track distance, and multiply our cross-track distance by this ratio. That sounds complicated, but it is not when you try a few practical examples. Let me walk you through one. If you’ve gone 30 miles and you’re 4 miles off track, the ratio of 60 to 30 is 2. Multiply your 4 miles of cross-track error by 2, and you get 8 — which is the equivalent error in 60 miles, and that gives you the 8 degree track angle error directly. That’s the whole trick — scale your cross-track error up or down to what it would be over 60 miles, and the number you get is your track error in degrees. That’s the core of the 1 in 60 rule and how to expand or contract the triangle to make it work at any along-track distance.