Posted by Blair on .
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:

Math 
Damon,
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 
Math 
Blair,
The answers are wrong for this one.

Math  sign wrong 
Damon,
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 
Math  sign wrong 
Blair,
The answer is still wrong :(

Math 
Damon,
check my arithmetic carefully