Taylor Series for sine using functions

Question

I've created a fatorial function that would help me calculate the taylor expansion of sine. There are no evident mistakes in my code, but the returned value is wrong. That's my code:

PI = 3.14159265358979323846

def fatorial(n):

    fatorial = 1
    for i in range(1,n+1,1):
        fatorial = fatorial * i
    return fatorial 

def seno(theta):
   
    n = 1
    k = 3
    eps = 10**-10
    theta = (theta*PI)/180
    x = ((-1)**n)*((theta)**k)
    y = fatorial(k)
    while x/y > eps or x/y < -eps:
        theta = theta + (x/y)
        n = n + 1
        k = k + 2
        x = ((-1)**n) * ((theta)**k)
        y = fatorial(k)
    return theta

Answer 1

You are summing theta in the while-loop and therefore using an adjusted angle for the next elements of the Taylor series. Just add a thetas (for the sum) like so (I have taken the liberty to add some performance improvements, no need to recalculate the full factorial, also calculated first elements explicitly and changed the limit check to avoid recalculations of x/y):

import math

PI = 3.14159265358979323846

def fatorial(n):
    fatorial = 1
    for i in range(1,n+1,1):
        fatorial = fatorial * i
    return fatorial 

def seno(theta): 
    n = 1
    k = 3
    #eps = 10**-10
    theta = (theta*PI)/180
    thetas = theta # <- added this
    x = -theta*theta*theta
    y = 6
    #while x/y > eps or x/y < -eps:
    while k < 14:
        thetas = thetas + (x/y) # sum thetas
        n = n + 1
        k = k + 2
        #x = ((-1)**n) * ((theta)**k)
        x = ((-1)**(n%2)) * ((theta)**k) # but use the original value for the series
        #y = fatorial(k)
        y *= k * (k-1)
    return thetas # return the sum

if __name__ == '__main__':
    print(seno(80), math.sin(8*PI/18))

this results in

0.984807753125684 0.984807753012208