import math
 
def main():
    x = input('Enter value to get sine of in radians: ')
    if 0 <= x <= 2*math.pi:
        n = input('Enter number of iterations for approximation: ')
        total = 0.0
        fac = 1
        xp = x
        sign = 1
        for i in range(1, 2*n, 2):
            total += sign * xp / fac
            xp *= x ** 2
            fac *= (i+1)*(i+2)
            sign *= -1
        real_sin = math.sin(x)
        print 'The approximated value of the sine of', x, 'is:', total
        print 'The actual value from math.sin is:', real_sin
        if real_sin != 0:
            print 'The relative error is:', abs(real_sin - total) / real_sin
    else:
        print 'x is an illegal value.'
    else:
        print 'The absolute error is:', abs(real_sin - total)
 
main()

import random
import math
 
def main():
    # You didn't know how to do the below loop for this problem set, but you
    # should know what it does now. It creates a list that contains all of
    # the number of iterations you were supposed to test. Try it!
    for n in [10, 100, 1000, 10000, 100000, 1000000]:
        total = 0
        for i in xrange(n):
            x, y = random.random(), random.random()
            if math.sqrt(x ** 2 + y ** 2) < 1.0:
                total += 1
        mypi = 4.0*total / n
        print 'Estimating pi with', n, 'iterations:', mypi
        print 'Value of math.pi is', math.pi
        print 'Error is', abs(math.pi - mypi) / math.pi
        print
 
main()