A program to define sine function

[Gribouillis]

You can improve the code by using the relation that exists between two consecutive terms of the series, thus avoiding the computation of high powers of x and factorials

from itertools import count, islice
import math

def sine_terms(x):
    t = x
    f = -x ** 2
    for n in count(3, 2):
        yield t
        t *= f /((n-1) * n)
        
def sine(x, n):
    return sum(islice(sine_terms(x), n))

if __name__ == '__main__':
    print(sine(6, 25), math.sin(6))

Output:
-0.2794154981989254 -0.27941549819892586

There is necessarily an error for large values of x because the sine function is not a polynomial. You could find online estimates for the remainder of the series, for example here is an estimate for the remainder of the exponential series, which should work also for the sine series (I haven't checked the estimate). In fact, for the sine series you can get an easy bound for the error by using the Lagrange remainder of the Taylor formula.