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Aren't cyclic phenomena interesting?

   Sin(0) = Sin(n * 360) = 0
   cos(0) = cos(n * 360) = 1

  Where n is an integer multiplier

If degrees equal to or greater then 360 then subtract
360 until degrees is less then 360.

That's why Compass cards don't indicate 360 and,
by analogy, why 24 hour clocks don't indicate 2400.

The confusion occurs because a circle is divided
into 360 angular parts but compass cards are zero based
and show 0..359 (still 360 graduations, 0 mark plus
359 other marks).  You steer 0 and not 360.


Maybe I should have stopped here.  If you don't want
to be totally confused, please exit here.


You are steering 90 deg and are asked to execute
a 450 deg turn starboard.  Don't scratch your head
questioning the sanity of the order.  Do it this
way:

Tun right and go around a full circle.  Then, continue
the turn until your compass indicates 180; Done.


Accumulated degrees turned:

    0   90  180  270  360  450

Magnetic compass:

   90  180  270   00   90  180


Integrating compass (perhaps gyro driven):

   90  180  270  360  450  540


Notice, the integrating compass does define 360 as a valid
direction. If you still think there is a paradox, yes, it
occurs when the integrating compass approaches full scale.


Watching your magnetic compass, when you have turned an
accumulated 270 deg, it will indicate

359.999999999999....   and finally go over to 0.

No, in the real world of magnetic compasses, you don't
get closer and closer to 360 but never get there.

Watching you integrating compass, you can always get a magnetic
reading by subtracting 360 deg until the result less than 360.

Question:

Isn't there an analogous situation for the cyclic behavior of
clocks as elapsed time accumulates?


                                         --- CHAS

   

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