Python's floats are emphatically not doing symbolic arithmetic - they use the platform's binary floating point facilities, which can only represent a subset of rationals exactly.  All other values are approximated.

In particular, this shows the exact value of the approximation used for pi:

>>> import math
>>> from decimal import Decimal
>>> Decimal(math.pi)
Decimal('3.141592653589793115997963468544185161590576171875')

That isn't the mathematical pi, but is the closest approximation representable as a platform Python binary float (same as type "double" in the platform C).

Here's the difference between math.pi and a better decimal approximation to pi:

>>> Decimal(math.pi) - Decimal("3.141592653589793238462643383279502884")
Decimal('-1.224646799147353177224094238E-16')

So math.pi is a little bit smaller than the mathematical pi.

Mathematically, using the subtraction formula for sine:
 
sin(pi - e) = sin(pi)*cos(e) - sin(e)*cos(pi) =
0*cos(e) - sin(e)*-1 =
0 + sin(e) =
sin(e)

So mathematically sin(pi - e) = sin(e), and if |e| is close to 0 then sin(e) ~= e.

For that reason, it's not a coincidence that the result you got for math.sin(math.pi) = 1.22464...e-16 is approximately equal to the difference (shown above) between the mathematical pi and math.pi.  It's not due to gross inaccuracy in sine, it's due to that pi isn't exactly representable as a Python float.

Note that this has nothing to do with Python specifically.  You'll see the same kind of behaviors in any language exposing the platform floating point facilities.

If you need to do symbolic computations, then you need much fancier facilities.