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I started by noticing that the derivative of $\sin(x)$ is always between $1$ and $-1$. Therefore if i have a line that intersects the $x$ axis at $45°$ it will aways pass through the line once or be tangent to it. In other words $\sin(x)=\pm x-a$ has only one solution for any $a$ and sign for $x$. So using the rotation matrix to rotate $\sin(x)$ $-45°$ around the origin I get $$\sin\left(\frac{x+y}{\sqrt{2}}\right)=\frac{y-x}{\sqrt{2}}$$ I got stuck on trying to write this as a function of $x$ but i do believe it can be. And as an add on question can the same be done for $\sin(x+y)=y-x$

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    $\begingroup$ if you have an program you can use implicitplot $\endgroup$ Jun 19, 2017 at 11:15
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    $\begingroup$ I have the plot up on demos and its a cool looking graph, like curvy stairs $\endgroup$
    – Ben Martin
    Jun 19, 2017 at 11:26

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Well, one way to approach this is to use the power series expansion around the origin of $$sin(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}- \cdots$$ A first approximation would then be $sin(x+y)=x+y-\frac{(x+y)^3}{3!}=y-x$, which yields $y=-x+\sqrt[3]{12x}$. This provides a nice plot (Desmos):

enter image description here

You could take more powers of the in the Taylor-expansion into account of course.

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    $\begingroup$ This is a cool approach and quickly reach better and better appoximation of a function of x (I've noticed that it can only be expressed like that in even terms otherwise y has multiple values) The issue is that you very quickly get very hard to solve polynomials. Would you know any way aronnd that ? $\endgroup$
    – Ben Martin
    Jun 19, 2017 at 21:42
  • $\begingroup$ You are right, for higher powers, it becomes difficult: this amounts to making $y$ explicit in the equation $(x+y)^5-20(x+y)^4+240x=0$. I tried to work with approximations of $(x+y)^k=x^k+ky$ when $y$ is very small. But this leads to functions that are not at all a good approximation of the original one. $\endgroup$ Jun 21, 2017 at 9:43

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