# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**360 degree slide rule trig**

**From:**Paul Hirose

**Date:**2016 Nov 15, 21:37 -0800

To continue the "Rhumb line by slide rule" thread I need to explain how to evaluate trig functions at any angle, not just 0 to 90 degrees. My initial thought was to explain cosine and sine as x and y Cartesian coordinates. However, angles on Cartesian diagrams are normally measured from zero at the east direction, increasing counterclockwise. That's completely different from navigational practice. For our purposes it's more convenient to think of sine and cosine as easting and northing. The sine of an angle is the easting per mile made good on that course. For example, sin 20° = .34, so each mile on course 020 equals .34 mile easting. The cosine of an angle is the northing per mile made good on that course. For example, cos 140 = -.77, so each mile on course 140 equals -.77 mile northing. I didn't use the usual terms "departure" and "difference of latitude" because the "sign" of those values is generally a compass direction, whereas in trig we must use plus or minus. To be proficient in slide rule trig you should know how to set and read any angle up to 360°. The hard way is the textbook method: set the slide rule aside and make an auxiliary computation to reduce the angle to its equivalent in the 0 - 90 range of the scales. The easy way is to set the angle directly on the rule. I'll show you how. From left to right on scale S observe sin 10°, sin 20°, etc., up to sin 90°. But the angles don't stop there. Now reverse direction. The sin 80 mark also stands for sin 100, sin 70 for sin 110, all the way to sin 10, which is equivalent to sin 170. For example, to set sin 156.1, start at sin 90 and count degrees from right to left: 100, 110, ... 150, 155, 156.1. For an angle greater than 180, begin another cycle at the left end of S. The sin 10° graduation is equivalent to sin 190°, 20° is sin 200°, up to sin 270° at the right. Finish the cycle with a trip from right to left, which ends at 360°. Not exactly. One problem is that sines of angles less than about 6° and cosines of angles greater than 84° are not on scale S. For these we use scale ST. For example, set sin 176. From left to right, the first cycle on S takes you to 90, then from right to left to 170 at the 10° mark. You can count left 4 more degrees, but 174 (at the 6° mark) is the last whole degree. So exit the left end of S and enter at the right end of ST. Continue leftward. The 5° graduation is equivalent to 175, and 4° is the same as 176. We have arrived. On C or D read sin 176° = .07. Is it positive or negative? Remember that sines are equivalent to easting. Course 176 takes you east, so sin 176° = +.07. If you continue left on ST, the 1° mark is equivalent to sin 179°. That's the last whole degree on ST. After you reach the left end of ST and reverse direction, the 1° mark stands for 181°, 2° = 182°, etc. After 185° you exit ST on the right and enter the left end of S. In this way ST provides the angles that aren't on S. Cosines are the same, except that the zero is at the right end of S. From there the angle increases leftward until you exit S and enter ST at about 84.5°. A vital thing to remember is that when you're beyond the "official" range of a scale, the numbers are ignored except to help count degrees as you move along the scale. A convenient fiction is to imagine S begins with 0 on the left. Ignore the ST scale unless you need to make the setting or reading there. Sines and cosines of negative angles are no problem. For example, what's the cosine of -30°? Visualize -30 (= 330) on a compass rose, and remember that cosines are equivalent to northing. Clearly, northing is the same whether you're on course 30 or -30. So cos -30 = cos 30. On the other hand, sin -30 is the negative of sin 30, since the respective eastings have identical magnitudes but opposite signs. So far I've discussed setting angles, but the principle of cycling back and forth also applies to the reading of angles (arc sine and arc cosine). For example, what's the arc sine of -.4? Well, sine is equivalent to easting, and course must be between 180 and 360 to get negative easting, so the answer is in that range. In fact, two different courses (symmetrical about 270) will produce the same easting. Set the cursor at .4 on C or D. Now observe scale S. From left to right it goes from 0 to 90, then the return trip from 90 to 180. That's the first cycle. For the second cycle, the 10° mark is equivalent to 190, 20 is equivalent to 200, and 3.6 more degrees takes you to the hairline. Arc sine -.4 = 203.6. That's the first answer. Continue right to 270 at the end of S. Reverse direction and count back to the cursor: 280, 290, ... 330, 335, 336.4. That's the other arc sine of -.4. I like to use a finger, not just my eyes, to go back and forth on a scale. That is, I point to one end and say "zero", point to the opposite end ("90"), back the other way ("180"), etc. It helps keep my mind on track. Later, when I explain how to get 360° of tangents, we'll see that one cycle back and forth on the T scale is 90°, vs. 180° on S. So any trick to avoid confusion helps.