One series for finding sin(x) is
sin(x) |-> x/1! - x3/3! + x5/5! - x7/7! +...
It is obvious from this that sin(0) = 0, since all the terms evaluate to 0.
If infinite series are not your thing, try this infinite product, although it is not very clear in HTML:
sin(x) = (1 - x2/π2
) * (1 - x2(22π2)) * (1 - x2/(32π2))* ...
Note that these definitions of sine require an input in radians, not degrees, with which some are more familiar. To convert to degrees, simply multiply by 180/pi.
Like cosine, sine is periodic in 2π. The derivative of sine is cosine, and its integral is -cosine.
Compare this series to the series for cosine and hyperbolic sine.