Natural Science Forum / Physics / General Physics / March 2007

No it can easily be said the opposite way also. The
first sentence I mean. It depends on where the absence
of precision is applied. To the plan of solution or to
the numbers involved. In what I said it applied to the
plan.  

***{Pi is not merely the ratio of the circumference of a circle to its
diameter, but also the ratio of half the circumference to the radius.
Your method uses a semicircle of radius 1, divides it into n/2 polygon
segments, drops a perpendicular from the center of the semicircle to the
midpoint of each segment, and takes the tangent of the resulting central
half-angle. (I am assuming the polygon segments are inscribed within the
semicircle, rather than being outside it, since that gives the most
accurate approximation.) Since the side opposite the angle is half the
length of a polygon segment, there are n of them in the semicircle, and
the angle is therefore 180/n degrees. Unfortunately, the side adjacent
to the angle is shorter than the radius, hence less than 1. Fortunately,
however, the adjacent side approaches 1 as the central half-angle
approaches zero, and so the method works when the tangent is used. But
the reason it works is that tan x approaches sin x as x approaches zero.
Thus a better approximation is:

n sin (180/n)

It is closer to the actual value of pi at all values of n than is n tan
(180/n), though at large values of n (above 10^5 on my calculator) the
differences cease to matter.

For example, if n = 6, then n tan (180/n) = 6 tan (180/6) = 3.464...,
whereas n sin (180/n) = 6 sin (180/6) = 3. Note that 3 is closer to
3.14159... than is 3.464...

Why does the sine work better than the tangent? Because the sine is the
ratio of the opposite side to the hypotenuse, and the hypotenuse is
always exactly equal to 1. The tangent is the opposite side divided by
the adjacent side, and, as noted earlier, the adjacent side is always
less than 1, though it converges on 1 as the central half-angle
approaches 0.

Result: the tangent works, but the sine works better.

This is a great way to get pi on a calculator that has no pi key, but
has trig functions. For example, 10^20 sin (180/10^20) = 3.14159265359,
which is all the significant digits my pocket calculator allows. It has
a pi function but it is no better, giving exactly the same number. (On
my calculator the smallest value of n that gives that result to 12
significant digits is 10^7.)

There are lots of examples where it is useful to be aware that as the
angle approaches zero, the sine and the tangent converge. In many of
them, the tangent may be used when the angle is very small, but the sine
must be used when it is large. Stellar aberration, for example, can just
as well be calculated using the tangent as the sine, in most cases,
because the angles tend to be very small. The aberration of downward
falling raindrops caused by the forward motion of an automobile, on the
other hand, is going to be a large angle, and calculating such things
requires use of the sine.

Very interesting stuff.

--Mitchell Jones}***

*****************************************************************
If I seem to be ignoring you, consider the possibility
that you are in my killfile. --MJ