Math
posted by Blair .
Use Newton's method to approximate a root of the equation 5sin(x)=x as follows.
Let x1=2 be the initial approximation.
The second approximation x2 is:
and the third approximation x3 is:

5 sin x = x
let y = x  5 sin x, search for y = 0
dy/dx = y' = 1  5 cos x
Xn+1 = Xn + y(Xn)/y'at Xn
X1 = 2
y = 2  5 sin 2 = 2  4.54 = 2.54
y'=1  5 cos 2 = 3.08
X2 = 2 2.54/3.08 = 1.17
y = 1.17  5 sin 1.17 = 3.43
y' = 1  5 cos 1.17 = .951
X3 = 1.17 3.43/.951 = 4.77
This is unlikely to work the way you want because you are jumping from cycle to cycle of the original sine wave 
The answers are wrong for this one.

sorry, sign wrong. I drew my picture wrong
5 sin x = x
let y = x  5 sin x, search for y = 0
dy/dx = y' = 1  5 cos x
Xn+1 = Xn  y(Xn)/y'at Xn
X1 = 2
y = 2  5 sin 2 = 2  4.54 = 2.54
y'=1  5 cos 2 = 3.08
X2 = 2 + 2.54/3.08 = 2.82
y = 2.82  5 sin 2.82 = 1.24
y' = 1  5 cos 2.82 = 5.74
X3 = 2.82 1.24/5.74 = 2.60 
The answer is still wrong :(

check my arithmetic carefully