As x gets closer and closer to zero, sin(x) approaches x in radians.
To convert from degrees to radians, one must multiply by pi/180.
Since .1111... represents 1/9, .555... represents 5/9.
Disregarding any factor of a power of ten, 1/5555... represents 1/(5/9), or 9/5, so as more 5's are strung together in the denominator, the closer we get to 9/5 degree divided by a larger and larger power of ten. (The 9/5 itself is a limit, or asymptote.)
So in the conversion to radians we get (9/5) * pi/180 = pi/100, or just pi divided by a power of ten, to be combined with the other powers of 10, so the leading nonzero digits approach those of pi.
Here are the decimal approximations of 1/5...5, with n 5's:
n approx.
1 0.20000000000000
2 0.01818181818182
3 0.00180180180180
4 0.00018001800180
5 0.00001800018000
6 0.00000180000180
7 0.00000018000002
8 0.00000001800000
9 0.00000000180000
10 0.00000000018000
so of course when multiplied by pi/180, the digits of pi show up.

Posted by Charlie
on 20130402 11:08:08 