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.
These are observations using the sequence generator on a TI-84 Plus CE. Some of the answers may be inaccurate due to rounding errors.
It was not stated whether the angles should be read as radians or degrees, but it appears not to matter.
In either case f and g seems to be heading towards -1 and 0 respectively. In radians it converges extremely quickly.
-1cos(0)-0sin(0)=-1
-1sin(0)+0sin(0)=0
In degrees it converges more slowly. f(64)=-.01 is the first negative, corresponding to g(64)=0.9999938 (the highest value of g.) From there they decrease and seem to head towards -1 and 0.
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Posted by Jer
on 2020-05-18 10:39:36 |
Observations
