# Fun with TI-83: billions upon billions of cosines

Okay, maybe not billions. But by taking cosines repeatedly, one can find the solution of the equation ${\cos x = x}$ with high precision in under a minute.

Step 1: Enter any number, for example 0, and press Enter.
Step 2: Enter cos(Ans) and press Enter

Step 3: Keep pushing Enter. (Unfortunately, press-and-hold-to-repeat does not work on TI-83). This will repeatedly execute the command cos(Ans).

After a few iterations, the numbers begin to settle down:

and eventually stabilize at 0.7390851332

Explanation: the graph of cosine meets the line ${y = x}$ at one point: this is a unique fixed point of the function ${f(x)=\cos x}$.

Since the derivative ${f'(x)=-\sin x}$ at the fixed point is less than 1 in absolute value, the fixed point is attracting.

Now try the same with the equation ${10 \cos x =x}$.

This time, the numbers flat out refuse to converge:

Explanation: the graph of ${f(x)=10\cos x}$ meets the line ${y = x}$ at seven point: thus, this function has seven fixed points.

And it so happens that ${|f'(x)|>1}$ at each of those fixed points. This makes them repelling. The sequence has nowhere to converge, because every candidate for the limit pushes it away. All that’s left to it is to jump chaotically around the interval ${[-10,10]}$. Here are the first 1024 terms, plotted with OpenOffice:

Clearly, the distribution of the sequence is not uniform. I divided the interval ${[-10,10]}$ into subintervals of length ${0.05}$ and counted the number of terms falling into each.

What is going on here? Stay tuned.

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