we are now going to look at the problem of calculating the distance on the earth between any two points at the same latitude but not necessarily at the equator we have come across calculation of earth distances before the great circle distance between any two points on the earth can be calculated using a general equation involving spherical trigonometry but this is beyond the scope of the syllabus consequently problems involving the calculation of great circle distances are limited to those where the two points lie on special great circles these are when the points are both on the same meridian or when they are both on the equator suppose we have a chart like this with a root on it we measure the track distance with a pair of dividers if there is no graduated scale line at the bottom of the chart we can measure against the number of minutes of change of latitude can we do the same thing against the change of longitude the answer is no of course one minute of change of longitude is not one nautical mile except at the equator let's see why we'll go back to some basic concepts all meridians along with their associated anti-meridians are great circles however only one parallel of latitude is which is the equator all other parallels of latitude are small circles one nautical mile equals one minute on any great circle for instance on a meridian or at the equator therefore at any latitude one minute of change of latitude always equals one nautical mile however this is true only for change of latitude or at the equator for a change in longitude so now let's define departure and work out how to calculate it departure is defined as the distance between two meridians along a specified parallel of latitude so departure is always a rum line the parallel of latitude it's important to understand this there will also be a great circle distance between two points which will be shorter but this is not departure it's also possible to calculate either the rum line or the great circle distance between two points not at the same latitude but these are not departure either it has to be along the parallel of latitude to meet the definition it's usually measured in nautical miles but it could be in kilometers departure will always be greatest at the equator where it is part of the great circle it reduces as the meridians converge and becomes zero at the pole where all meridians meet at the equator the change of longitude in minutes is multiplied by one at the pole the change of longitude in minutes is multiplied by zero departure is therefore a function of cosine which has a value of 1 at 0 degrees and 0 at 90 degrees the equation is therefore that departure equals the change of longitude in minutes multiplied by the cosine of the latitude in the same way that we examined the sine curve when looking at convergency let's look at some of the more common values of cosines of angles because these frequently come up in questions on general navigation here is a cosine curve let's look at the angles between zero and ninety degrees the cosine of zero degrees is one the cosine of thirty degrees is point eight six the cosine of 45 degrees is 0.7071 the cosine of 60 degrees is a half or 0.5 the cosine of 75 degrees is about a quarter or more accurately 0.259 and the cosine of 90 is zero reduces from 1 to 0 as the angle increases between 0 and 90 degrees the relationship between departure and change of longitude is therefore defined by the cosine of the latitude this gives us the basic departure formula departure equals the change of longitude in minutes multiplied by the cosine of the latitude consider two meridians on the earth joined by a parallel of latitude we can extract the basic elements of this situation in the sketch in this example the change of longitude is 20 degrees multiply by 60 to convert it into minutes then multiply by the cosine of 52 degrees this gives a departure of 738.8 nautical miles test questions are unlikely to be as simple as that they usually involve conversion of units from kilometers to nautical miles or some rearrangement of the equation take this example an aircraft at position 60 north and 5 degrees and 22 minutes west flies 165 kilometers due east what's the new position firstly convert 165 kilometers to nautical miles to do this divide 165 kilometers by 1.852 the answer is 89 nautical miles now we need to substitute into the departure formula so departure equals the change of longitude in minutes multiplied by the cosine of the latitude which of these do you know the departure is 89 nautical miles substitute it into the equation the latitude is 60 north its cosine is a half rearrange the equation this gives 178 minutes of change of longitude this is 2 degrees and 58 minutes from an initial longitude of 5 degrees and 22 minutes west so since we are going eastwards it is 5 degrees 22 minutes west minus 2 degrees 58 minutes which equals a new longitude of 002 degrees 24 minutes west the new position is 60 north and 2 degrees and 24 minutes west here is another example of a different substitution in this one you are given the departure and have to find the latitude instead in which latitude is a difference of 44 degrees and 11 minutes equivalent to a departure of 2000 nautical miles as usual start with the formula departure equals the change of longitude in minutes multiplied by the cosine of the latitude which of these do you know you know the departure and the change in longitude write them in we now need to convert 44 degrees and 11 minutes into minutes it comes to 2651 minutes now rearrange the equation to make the cosine of the latitude the subject use the arc cosine function on your calculator to solve for the cosine which gives us the latitude of 41 degrees so the latitude is 41 degrees remember it could be north or south convergency and departure are symmetrical either side of the equator you need to be able to solve questions where you are given departure at one latitude and have to solve it for another consider this question an aircraft leaves position g at latitude 40 south and flies the following rum line tracks and distances g to h track 180 true distance 240 nautical miles h to j track 270 true distance 240 nautical miles j2k track 0 0 0 true distance 240 nautical miles what is the rumline bearing and distance from k to g there are two ways you can solve these types of questions one is by a double substitution into the basic departure formula it works but it's slower the other is by use of the following formula departure at latitude a divided by departure at latitude b equals cosine a over cosine b the new formula is quicker let's do the question both ways firstly by using a double substitution into the basic departure formula and then secondly by using the new method you should then see the advantage of the new formula