In navigation, trigonometry is a highly useful discipline; estimating set and drift of tides, courses to steer to make good a target, true winds from apparent winds, even tide heights, are all exercises in plane trigonometry.
The problem, of course, is that few people have either the ability or the desire to memorize a trig table. Fortunately, there is no reason to, at least for those cases where what one desires is a quick, “dirty” answer. With some practice, one can easily determine answers to acceptable precision (say, ±10%) using mental arithmetic and the three trig tricks presented below. These tricks may seem overly quantitative to some, or imprecise to others. However, the advantages while sailing or racing are many.
All three of the shortcuts are based on the mathematical properties of the trig functions, principally sine and cosine. Rule one allows easy calculation of the sine; rule two easy calculation of the cosine; and rule three eliminates ever having to calculate either sine or cosine for more than six angles. In all three cases, my goal is to be precise to better than 10%, which is certainly acceptable for mental arithmetic. If more precision is needed, the quick answer provided here will probably satisfy any pressing requirements while you spin the maneuvering board or crunch the calculator. Those readers familiar with trig may cringe at some of the shortcuts, but all answers will be within the magic 10% error, our goal.Rule one:
The sine of an angle equals the angle measured in radians, up to angles of 50°. Radians are an alternative, and more natural, way of measuring angles. In any circle, an arc as long as the circle’s radius maps out one radian, which equals approximately 57°. To find the sine of an angle, divide the angle by the number of degrees per radian, or 60. (Use 60 rather than 57 because the division is easiermost navigators seem pretty proficient at dividing by 60 anyway.) For example, by this method the sine of 15° is 15/60 = 0.25. The actual value is 0.26, for an error of 4%. This method is in error by more than 10% for angles of 50° to 90°, but this isn’t a problem either: remember that sin 60° = 0.9, and the sine of anything higher than 70° is 1. While this list may seem arbitrary, rule three will show that one only needs to know the sine of only six angles to figure out the rest.Rule two:
The cosine of an angle is the same as the sine of the angle’s complement. (Brings you back to that high school geometry class, doesn’t it?) An angle and its complement sum to 90°. Thus Cos 20° = Sin 70°. This rule is good for all angles, and is exact. Another (though less convenient) rule for the coSine is below. It is significantly less simple than the corresponding rule for the Sine, and isn’t really needed once you’ve mastered rules one and three.Rule three:
Interpolate whenever possible, preferably by using angles easily divisible by 60. For example, to determine Sin 22°, I would go through the following process: Sin 15/60 = 0.25, Sin 30/60 = 0.5, so Sin 22° would be in the middle, or 0.375. The actual value is 0.3746. Interpolation is a learned skill, but one in which most navigators are well practiced.
We’ll close with two examples of the rules in action. Case one: On course 000°, speed 6 knots, determine speed over the ground with tide setting 135°, drifting 2 knots. Speed along the course will be reduced by the drift multiplied by the coSine of the angle between the course and drift. First, determine the coSine of the angle: Cos 45° = Sin 90°- 45° = Sin 45° = 45°/60 = 0.75. So, our speed made good is (6 – 2 x 0.75) knots = 4.5 knots. The exact result is 4.6 knots, for an error of 6%.
Case two: How much northing and easting will a vessel on course 033°, speed 5 knots make in one hour? The northing is equal to the total distance traveled multiplied by the Sine of 33°. Approximate Sin 33° as one fifth of the way between Sin 30° and Sin 45°: Sin 30° = 30°/60 = 0.50 Sin 45° = 45°/60 = 0.75 thus, Sin 33° = 0.55.
Northing is then 5 miles x 0.55 = 2.75 miles. Easting is total distance traveled multiplied by the coSine of 33°. Cos 33° = Sin 57°, which is close enough to Sin 60° = 0.9 (rule three) for me. Thus, easting is 5 miles x 0.9 = 4.5 miles. Actual values are 2.7 and 4.2 miles, so we’re within 1% and 7%.
These rules allow you to perform first-level navigation calculations while you’re in the cockpit, the head, or eating dinner. Equally important, they provide a rapid way of getting an approximate answer to a calculation as a check of more complicated procedures, or answers provided by those black boxes on so many navigation tables. Do not rely only on these mental calculations, though. As always, the prudent navigator uses more than one tool to guide his or her boat through the water. With practice, these calculations become automatic, as will the savings in time and effort their use will provide.
Larry McKenna is an assistant professor of geology at the University of Kansas and a navigation enthusiast.
