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.
Plot f(n) and g(n), with g(n) as the x coordinate and f(n) as the y coordinate. All values lie on a unit circle centered at (0,0). The sequence of points moves clockwise around the unit circle. When f(n) and g(n) are both positive, by inspection f(n) is decreasing and g(n) is increasing. Eventually, f(n) turns negative, at which point f(n) continues decreasing and g(n) starts decreasing. The point moves clockwise around the unit circle more and more slowly as it approaches but never reaches its limiting values of (-1,0).
Geometric Interpretation
