Is there any sensitive equation solver which will not show the result as approximately $0$ for this equation:
$$\cos x=e^{-\Large\frac{3}{10^{45}}}$$
or how can I calculate it?
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Mathematica can probably do it for you. I must ask, why isn't 0 a sufficiently accurate answer? – Mathily Oct 27 '16 at 18:00
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Need for a physical calculation – user383233 Oct 27 '16 at 18:03
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PARI/GP is already powerful enough (even with standard precision) to calculate this. – Peter Oct 27 '16 at 18:35
3 Answers
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An excellent approximation for $$\arccos(e^{-x})$$ for $x\approx 0$ is $$\sqrt{2x}$$ (In the case you have to calculate such things in the future only having access to a normal calculator)
In your example, the difference between the values is about $4\cdot 10^{-68}$. Even, if $x$ is , for example $0.001$ , the error is less than $10^{-5}$.
Peter
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Where does this approximation come from? I've never run across it, and it seems rather powerful. – zz20s Oct 27 '16 at 19:01
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Look here, this might be an explanation : https://www.wolframalpha.com/input/?i=cos(sqrt(2x))-exp(-x) – Peter Oct 27 '16 at 19:27
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When you want to solve $x$:
$$\cos(x)=\text{K}\Longleftrightarrow x=2\pi\text{n}\pm\arccos\left(\text{K}\right)$$
Where $\text{n}\in\mathbb{Z}$
So, when $\text{K}=e^{-\frac{3}{10^{45}}}$:
$$\cos(x)=e^{-\frac{3}{10^{45}}}\Longleftrightarrow x=2\pi\text{n}\pm\arccos\left(e^{-\frac{3}{10^{45}}}\right)$$
Jan Eerland
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If you want a numerical approximation like you asked, SageMath can do it for you.
You will find:
$$x\approx 7.7459666924148337703585307995647992216658\times 10^{-23}.$$
E. Joseph
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