One of the most common algorithms for this is to take first few terms of Taylor series for sine.
sin x = x - x3/3! + x5/5! - x7/7! + ...,
The more terms you take the better approximation you get. X is in radians here, but you can get radians from degrees quite easily. And then, as you see, only basic operations: +, -, *, / can be used to calculate the sine.
In machines with floating-point coprocessor chips CORDIC algorithm (with a few other modules) is used, as it also can be implemented in hardware.