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Why can't we write a simple equation where if we give the value of $x$ as input, we get the value of $\sin(x)$ as output?

By simple, I mean an equation involving just addition, division, subtraction and multiplication and exponentiation and keeping it in the realm of real numbers.

And I'm not necessarily asking for an equation, I'm asking if one does not exist, why is it so?

Even an intuitive explanation would work.

(It is my first question here so sorry for not being rigorous enough)

Pedro
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John_Nash
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  • Hello and welcome to math.stackexchange. Please be more specific: What do you mean by a "simple equation"? – Hans Engler Jul 12 '17 at 19:15
  • Probably not as "simple" as you'd want it, but this can be done to some degree with taylor series – Theo C. Jul 12 '17 at 19:16
  • "Simple" is in the eye of the beholder. What do you mean by "simple" here? – B. Goddard Jul 12 '17 at 19:16
  • You can, quite easily: $f(x)=\sin x$. You are probably referring to formulas that are built using the four basic arithmetic operations (addition, subtraction, multiplication, and division) and radicals. If you include exponentiation as an operation and allow complex variables, then $\sin(x)=(e^{ix}-e^{-ix})/(2i)$ is such a formula. If you allow taking limits, then of course the Taylor series works as well. – anon Jul 12 '17 at 19:16
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    If you don't allow complex variables or limits, then it is not possible to express $\sin$ in terms of the other types of elementary functions. One explanation of this that can presumably be turned into a proof (perhaps by examining asymptotic behavior) is that formulas built from these types of operations will never be periodic like sine is. – anon Jul 12 '17 at 19:18
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    With simple I was referring to an equation that does not extend infinitely. And I'm not just asking for an equation, I'm asking if it doesn't exist why is it so? – John_Nash Jul 12 '17 at 19:20
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    There is an equation, $f(x)=\sin(x)$. So you should be specific about what you are allowing or not allowing. I presume you are only allowing compositions of rational functions, radicals, and real number exponentiation. Are you looking for a formal proof or an intuitive explanation? I gave what I consider to be the latter that can be turned into the former. – anon Jul 12 '17 at 19:22

2 Answers2

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Because $\sin(x)$ is not an algebraic function. It is instead a transcendental function.

Here there is a proof.

Pedro
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It can $sin(x)=\sum^{\infty}_{k=0}\frac{-(1)^kx^{2k+1}}{2k+1}$, for further information look at Taylor Series

Daniel
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  • If we would only allow addition, multiplication, division and subtraction, without using complex numbers, why don't we get an equation in that case? – John_Nash Jul 12 '17 at 19:23
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    @John_Nash A rational function cannot be periodic. – anon Jul 12 '17 at 19:24