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

**Re: 360 degree slide rule trig**

**From:**Paul Hirose

**Date:**2016 Nov 27, 22:04 -0800

My previous message explained how to get 360 degrees of sine and cosine on the slide rule. http://fer3.com/arc/m2.aspx/360-degree-slide-rule-trig-Hirose-nov-2016-g37154 Now let's look at tangent and cotangent. Some slide rules have a double T scale, but I'll assume you have the more common single scale. It can do anything a double scale can do, though not always as conveniently. The numbers on T go from about 5.7° to 45° from left to right in black numbers (for tangents), and 45 to 84.3 from right to left in red (cotangents). Tangents and cotangents from T are between 0.1 and 1.0. Since tangents and cotangents are reciprocals, multiplication by black is identical to division by red. Example: an equation requires multiplication by tan 80° (black 80), which doesn't exist on T, so divide by cot 80° (red 80). Thus the range of T is effectively 5.7° to 84.3° in tan and cot. A similar technique finds the arc tangent or arc cotangent of a value greater than 1. Set a C index to the value on D, set the cursor to a D index, read angle on the opposite color on T. Example: to find arc tan 5.0, set cursor to right D index, set left C index to 5.0 on D, read 78.7° on red T. That illustrates the principle of reciprocal indices: at any position of the slide, the values at the C and D indices are reciprocals of each other. Scale ST is for angles beyond the left end of T: black 0 to 5.7 and red 84.3 to 90. Each black number on ST has a red counterpart, though the red numbers are usually omitted since the values are obvious: black 5 is also red 85, etc. Tangents and cotangents from ST are between .01 and .1. Examples: Cot 89 is at the black 1 (= red 89) graduation. Read .0175. Tan 88 (= black 88) is not on ST, but the scale has black 2 (= red 88). Set cursor to a D index, set black 2 to cursor, read tan 88 = 2.86 at C index. I.e., tan 88 is the reciprocal of cot 88. If the value is in a computation, say 2 / tan 88, work it as 2 * red 88. The left end of ST is about .57°. For smaller angles, mentally divide the black numbers by a power of 10. That is, the black 5 graduation also stands for black .5, black .05, etc. Corresponding red numbers are 85, 89.5, and 89.95. (For each power of 10 in the red numbers, insert a 9 before the units digit.) Adjust the result by the same power of 10. Examples: Tan .3° is off the scale, so read tan 3° = .0524 and divide by 10. Tan 89.5 (= black 89.5 or red .5) is off scale, so read black 5 (red 85) as black .5 (red 89.5). Set cursor to D index, set red 89.5 to cursor, read 115 at C index. Arc tan 5000. Since tan and cot on ST are all less than .1, first take the reciprocal: set a C index to 5000 on D, set the cursor to the D index. On C the cursor is at .0002. That's the cot. Read arc cot at black 1.15 (red 88.85). The red angle is arc cot .02, but the actual cotangent is two orders of magnitude less, so insert two 9s before the units digit of 88.85 and obtain 89.9885°. Another method is to read the black number: 1.15°. Adjust by two orders of magnitude to .0115°. Complement that by inspection to obtain 89.9885°. Those tricks will get tangents as close to 0 or 90 as desired. To exceed 90 and go all the way around the circle, cycle back and forth on T. It's the same method I explained for scale S, except that each cycle from left to right and back is 90°, not 180°. The first cycle is 0 to 90, the second is 90 to 180, etc. Use ST for the angles that run off the left end of T. To know where its value is negative, think of tangent as easting divided by northing. For example, on course 150 northing is negative, and on course 250 both easting and northing are negative. Thus tan 150 is negative and tan 250 is positive because the two negatives cancel. The easting divided by northing analogy also makes clear what angles have red numbers, i.e., when the scale yields cotangents. On any course from 45 to 135, easting exceeds northing (if you disregard any negative sign). Thus the quotient exceeds 1 and the angle is red. Likewise the quadrant from 225 to 315 is red. I call these the east and west quadrants since they're centered on 90 and 270. With the sign and color known, you can get a tangent of any angle without auxiliary computation. Examples: Tan 150 is negative and black. The first cycle on T goes from 0 to 90. From left to right the second cycle goes to 135. Then from right to left, the black 40 graduation is 140 and black 30 is 150. Read about -.58 as tan 150. Tan 250 is positive since easting and northing are both negative on that course. And it's red since the angle is in the west quadrant. The applicable T cycle begins with 180 on the left. The right end of T (black 45) is 225. Moving left, black 40 is 230, and the black 20 graduation stands for 250. To read the tangent, set the cursor to a D index, set black 20 on T to the cursor, read tan 250 = about 2.75 at the slide index. Note that 250° is red, i.e., the result from T is a cotangent. Yet I said to set the cursor to black 20. It would have been equally correct to say red 70. When you're outside the normal range of the scales, the numbers are only an aid to counting degrees. Use either color. Tan 91 is in the southeast quadrant so tangent is negative, and in the east quadrant so it's a red angle. It's in the tangent cycle that begins with 90. The first and last 5.7° of every cycle are on ST, so look there. Black 1 is equivalent to red 91, black 2 is red 92, etc. Set the cursor to a D index, set black 1 on ST to the cursor, read -57.3 on D. Tan 359.975 is in the second half of the cycle that begins at 270, so angle increases from right to left. Remember the technique of inserting 9s before the units digit. It works in reverse too. Remove two 9s from 359.975 to obtain 357.5. Start at black 5 as 355°, black 4 is 356°, black 3 is 357°. Then go left .5° more to black 2.5, or 357.5. The literal reading of the tangent is .0436. However, we adjusted by two orders of magnitude, so the true reading is .000436. And finally, the angle is in the northwest quadrant so the tangent is negative. An alternate method is to take the explement (= 360 - 359.975). The result is .025°. Read tan 2.5° with ST, adjust by two orders of magnitude and apply the negative sign. In the final part of this series I'll go into the coordinate transformation between rectangular (easting and northing) and polar (course and distance).