#Euler and Runge-Kutta methods are used to calculate sine from its derivative, cosine.
'''
Sine function is calculated using its derivative, cosine.
Using Euler and RK4 method, with
'''
import math

def rk4(x, y, yprime, dx = 0.01):   # x, y , derivative, stepsize
	k1 = dx * yprime(x)
	k2 = dx * yprime(x + dx/2.0)
	k3 = dx * yprime(x + dx/2.0)
	k4 = dx * yprime(x + dx)
	return y + ( k1/6 + k2/3 + k3/3 + k4/6 )

h = 0.01    # stepsize
x = 0.0     # initail values
y = 0.0     # for Euler
z = 0.0     # for RK4

while x < math.pi:
     print( x, y - math.sin(x), z - math.sin(x) ) # errors
     y = y + h * math.cos(x)       # Euler method
     z = rk4(x,z,math.cos,h)       # Runge-Kutta method
     x = x + h