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    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
    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.

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