Archimedes computed the circumference of the circle
by squeezing the circle between two polygons with n sides
obtaining so an approximations of the circumference and especially
an approximation of pi. Lets look at his calculation with
modern eyes: Writing x=pi/n, the length of the inner polygon is
2n sin(x), the length of the outer polygon is
2n tan(x) and the length of the circle is 2n x. We get the estimate
2n sin(x) ≤ 2n x ≤ 2 n tan(x) .
Dividing this by 2n/sin(x) gives
1 ≤ x/sin(x) ≤ 1/cos(x)
which is equivalent to
cos(x) ≤ sin(x)/x ≤ 1 ,
from which one can see the fundamental theorem of trigonometry
sin(x)
lim -------- = 1
x
We see that Archimedes picture of approximating the length of the circle
from inside and outside by polygons leads to an important result in
calculus. The result is important because it allows to get the formulas
for the derivatives of the trigonometric functions sin(x) and cos(x)
using addition formulas and double angle formulas:
(1 - cos(x)) = 4 sin2(x/2)
lim ------------ lim ---------------------- = 0
x x
Using this and the fundamental theorem again the limit h to zero can be taken:
sin(x+h)-sin(x) = sin(x) cos(h) + cos(x) sin(h)-sin(x)
---------------- ------------------------------------
h h
= sin(x) [ cos(h)-1 ] + cos(x) sin(h)
----------------------------------- -> cos(x)
h
In the same way, we get
cos(x+h)-cos(x) = cos(x) cos(h) - sin(x) sin(h)-cos(x)
---------------- ------------------------------------
h h
= cos(x) [ cos(h)-1 ] - sin(x) sin(h)
----------------------------------- -> -sin(x)
h
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