I noticed when playing with the calculator that if I repeatedly press "cos", starting with any number, it converges to $0.99984742$. Why does it happens??
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Welcome to math stack exchange! – Peter Oct 02 '16 at 20:55
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This question has been asked more than once. See for example (http://math.stackexchange.com/q/46934) – Jean Marie Oct 02 '16 at 21:51
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The equation $$\text{cos}(x)=x$$ can be solved via the iteration $x_{n+1}=\text{cos}(x_n)$ for any start value $x_0$ because after at most $2$ iterations, we have an absolute value less than $1$ and in the interval $[-1,1]$ , the derivate of $\cos(x)$, namely $-\text{sin}(x)$ has absolute value less than $1$.
You get the fixpoint of $\text{cos}(x)$ , which is the (unique) real solution of $\text{cos}(x)=x$.
You calculated in degrees and solved $x=\cos(x°)$. In radians, the equation would be $$cos\left(\frac{\pi x}{180}\right)=x$$ , which has a unique solution as well. The convergence is very fast in this case because the derivate of $\text{cos}(\frac{\pi x}{180})$ is very small.
In radians, the solution is $0.73908513\cdots $
the_candyman
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Peter
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The iteration works under some conditions shown in the Banach-fixpoint-theorem. – Peter Oct 02 '16 at 20:49
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If you had used the iterated solution to obtain this way $ \cos x = x $ in radians you get:
$$ x= 0.73908513 $$
Your calculator setting in degrees produces this solution:
$$ x= 0.9998474...$$
This is a way to evaluate fixed points convergence. A sketch would be helpful to see how the small squares shrink to the fixed point in x-y plot.
Narasimham
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