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I created the following on Desmos recently:
$y=\displaystyle\sum_{n=1}^{50}\sin\left(nx\right)$

While looking at it I noticed that it looks similar to $tan(x)$, so I tried to approximate it: $\sin\left(0.5\right)\left(\tan\left(-0.25x\right)\right)$

My question is why these numbers, and how can I be more accurate when plotting patterns in graphs?

Screenshot of graph
Link to Desmos Graph

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  • $\begingroup$ It in fact resembles $\;\cot x\;$ ... $\endgroup$ – DonAntonio Nov 18 '19 at 9:49
  • $\begingroup$ I found it to be very similar to $0.5\cot\left(0.25x\right)$. Do you have any idea as to why? Thanks $\endgroup$ – user726468 Nov 18 '19 at 9:53
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Well, we know that

$$\sum_{k=1}^{50}\sin kx=\frac{\sin\left(\frac{51}2x\right)}{\sin\frac x2}\cdot\sin(25x)$$

But $\;\sin a\sin b=\cfrac12\left(\cos(a-b)-\cos(a+b)\right)\;$, so the sum above is in fact

$$\frac{\cos\frac x2-\cos\left(\frac{101}2x\right)}{2\sin\frac x2}$$

and we already have there out cotangent and some other factor that I suppose is more or less small...

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