Departure — Page 266, Lesson 237

Let’s pick this up right where the worked example left off. We’ve just solved Example 3, which asked for the rhumb line bearing and distance of K from G, and we did it two ways. Now I want to show you the second method properly, because it’s the one you’ll actually use in the exam — it’s faster and it saves you time. Here’s the key formula we introduced: Departure at Latitude A divided by cos A equals Departure at Latitude B divided by cos B. That’s the relationship. But for solving, we rearrange it into a more useful form: Departure at Latitude A = Departure at Latitude B × (cos A ÷ cos B). Now, the trick is choosing which latitude to call B. You want B to be the latitude for which you already know the departure. In Example 3, we knew the departure at 44°S was 240 NM — that came from the 240 NM flown south along the meridian, which changed latitude by 4°. So we set B = 44°S, with Departure at B = 240 NM. Then we solve for A = 40°S, the latitude we want. So the calculation becomes: Departure at 40°S = 240 × (cos 40° ÷ cos 44°). Work that out and you get 255.6 NM — exactly the same answer we got with the double substitution method. And the rhumb line bearing is 270°(T), because the whole path was symmetric: you went south, then west, then north, so K ends up due west of G. Now, notice something important. The departure formula — Departure (NM) = change of longitude in minutes × cos latitude — is the basic tool. But this new ratio formula lets you skip a step. Instead of computing the change of longitude first and then re-substituting, you go straight from one departure to another using only the cosines of the two latitudes. That’s the time saver. Let me walk you through the logic once more, because it’s the heart of this section. Departure is the east–west distance along a parallel of latitude. It varies with latitude because the meridians converge toward the poles. At the equator, the meridians are farthest apart, so departure is at its maximum. At the poles, the meridians meet, so departure is zero. And in between, departure varies as the cosine of the latitude. That’s why the formula uses cos A and cos B — the cosine captures exactly how the east–west distance shrinks as you move away from the equator. So when you know the departure at one latitude, you can find it at another by scaling with the ratio of the cosines. That’s the whole point of Method 2. Now, after Example 3, the book gives you a set of practice questions. These are the book’s practice questions — let’s try them one at a time.