What is the code for this in Matlab or Mathematica using Newton's method?

Newtons Method boils down to

x_new = x_old - f(x_old)/ f ' (x_old)

where the function f(x)=0

f(x) = x-0.08 -.02 sin x
f ' (x) = 1-0.02 cos x
Now we just need an initial guess ..I suggest x=0
since we are repeating (iterating) the above calculation it suggests a "while" function being used until a certain criteria is met ie perhaps abs[x_new - x_old] <= 0.0001 ?

of course if you want to show off , you could include a check that f '(x_old) didnt equal 0 too

easier reply would be
f[x_]:=x-.08-.02Sin[x];
FixedPoint[#-f[#]/f'[#]&,0]...Show more