Let’s pick up with the 1 in 60 rule — Page 189, Lesson 170

Let’s pick up with the 1 in 60 rule. I’ve just shown you that if you take a right-angled triangle where the adjacent side is 60 metres long, then for small angles, the opposite side in metres is almost exactly equal to the angle in degrees. Now I want to expand that adjacent side to 60 metres in length — that means I have to multiply all the opposite-side values by 60 as well. Here’s the table I’m building. For angle z in degrees, I list tan z, then 60 times tan z. For z equals 1 degree, tan z is 0.017, so 60 tan z is 1.02. For 2 degrees, tan z is 0.035, 60 tan z is 2.1. For 5 degrees, tan z is 0.087, 60 tan z is 5.22. For 10 degrees, tan z is 0.176, 60 tan z is 10.56. For 15 degrees, tan z is 0.268, 60 tan z is 16.08. For 20 degrees, tan z is 0.364, 60 tan z is 21.84. Look at that pattern. There is a very close correlation between the angle z and 60 tan z. A one-degree track difference gives 1.02 nautical miles off track in 60 nautical miles along track. A ten-degree track difference gives 10.56 nautical miles off track in 60 nautical miles along track. Even at 15 and 20 degrees, the relationship is pretty close. But in practice, 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. Now, why doesn’t the rule give a perfect one-to-one correlation as the angles get bigger? There are two reasons. The first reason 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. The second reason is that the perfect correlation comes from using the arc of a circle. The explanation I just gave considers a right-angled triangle. The tangent of an angle is not a linear relationship, but up to about 20 degrees it is very close. However, as I said, you do not 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. Now let’s talk about expanding or contracting the triangle. Look at the diagram in Figure 10.6. The adjacent side represents 60 nautical miles 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, which 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 a figure other than 60. The problem is a simple one — a matter of similar triangles. Suppose we have only gone 30 miles along-track. For the same track angle error, 8 degrees left, we will only be 4 miles left of track. Look at Figure 10.7. 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. Four miles in 30 is obviously the same as 8 in 60, so the angle is 8 degrees. Ten miles in 120 is obviously the same as 5 in 60, so the angle is 5 degrees, and so on.