import math

def estimate_taylor_series(a_k, x, precision=1E-11):
    max_k = 0
    current = math.fabs(a_k(x, max_k))
    while current > precision:
        max_k += 1
        current = math.fabs(a_k(x, max_k))

    running_sum = 0
    for k in range(max_k):
        running_sum += a_k(x, k)

    return running_sum

def estimate_cos(x, precision=1E-11):
    return estimate_taylor_series(
      lambda x, k: ((-1)**k / math.factorial(2*k)) * (x) ** (2 * k),
      x,
      precision)

def estimate_sin(x, precision=1E-11):
    return estimate_taylor_series(
        lambda x, k: ((-1)**k / math.factorial(2*k+1)) * (x) ** (2*k + 1),
        x,
        precision)

print("Comparison: ")
print("estimated cos:",estimate_cos(3.04))
print("calculated cos:",math.cos(3.04))

print("estimated sin:", estimate_sin(3.04))
print("calculated sin:", math.sin(3.04))

# expected:

# Comparison:
# estimated cos: -0.9948439033600767
# calculated cos: -0.9948439033594595
# estimated sin: 0.1014179863214654
# calculated sin: 0.10141798631660187