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 non-zero 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.