# Newton's method to approximate a solution of f(x)=0 given an initial guess, x0.
# The user input is between the hashtags. You can run the code by cutting and 
# pasting into a SageMath Cell.
#
def f(x):
   return x - math.cos(x)

def fp(x):
   return 1.0 + math.sin(x)

x0 = 1
N = 10
#

x = x0
print(0, x)
for i in range(1, N+1):
   x = x - f(x)/fp(x)
   print(i, x)