Functions f:N→R, g:N→R are such that
f(n+1)=f(n)cos(g(n))−g(n)sin(g(n))
g(n+1)=f(n)sin(g(n))+g(n)cos(g(n))
If f(1)=0.8 and g(1)=0.6, find the limit of f(n) as n tends towards infinity.
I have just noticed that f(1)^2 + g(1)^2 = 1
Also, f(n+1)^2 + g(n+1)^2 = f(n)^2 + g(n)^2
So, by induction, f(n)^2 + g(n)^2 = 1 for all n.
In other words, they are points on a unit circle.
We have already determined (see earlier post) that if there is a limit, the limit of g(n) =0. It follows that if there is a limit, the limit of f(n) is either 1 or -1. More insight is needed to determine which one is the limit.
I think there is a geometrical interpretation of f(n) and g(n) that will tell us if there is a limit and whether it is (-1,0) or (1,0). Stay tuned for one more post.
Edited on May 20, 2020, 7:24 am
Aha!
