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GraphIf (for say, visual aesthetics) I would like to plot the graph of a sine wave on a 2D graph that intersects the $x$ axis with value of $y =0$ when $x$ is some arbitrary integer - how would I go about doing that?

Obviously it is trivial to find this function for the initial intersection; I'd like to investigate whether there exists a method to generate a wave that crosses this point after its first period.

To make it clear: I know how to plot the graph, but not how to find the seed value for the function.

Edit - added image: Apologies, I couldn't find a way to word the question for the actual outcome I wanted so tried to get a foothold by finding the intersect as $y =0$. I'm interested in getting all waves that 'peak' at a given value for $x$. I'm struggling a little with the terminology.

davbryn
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  • harmonic functions (e.g. (co)sine) have three parameters: frequency, phase and amplitude, for what conditions they must satisfy in your request? – Andrew Feb 27 '16 at 08:59
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    Here is a MathJax tutorial, which is how the math on this site is rendered so beautifully :) – Bobson Dugnutt Feb 27 '16 at 09:02
  • Have you heard about Fourier Transforms? You can use them to fit to pretty much any function, and they are build with sines and cosines. – Bobson Dugnutt Feb 27 '16 at 09:06
  • What do you mean by "crossing this point after the first period" ? Once the curve has passed this point, it will never be back there again...? – Jean Marie Feb 27 '16 at 09:17
  • It's not clear to us what you are looking for. The function $$y(t)=A \sin(\omega(t-x))$$ is a sine wave of amplitude $A$, frequency $f=\omega/(2\pi)$ and satisfies $y(x)=0$. But what exactly do you want? – Jyrki Lahtonen Feb 27 '16 at 09:43
  • I'm sorry, I'm not explaining this properly. I'll update my question with an image in a second – davbryn Feb 27 '16 at 10:13
  • There will be infinitely-many waves that peak at a given $x$ (turn the knobs on frequency and amplitude). Still not understanding what it is you're looking for. – AndJM Feb 27 '16 at 12:54
  • @sasha I'm interested only in integer n solutions for y = sin(x*PI /n) – davbryn Feb 27 '16 at 12:56

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If you fix the amplitude and phase, then the waves that first peak at $x = x_0$ are given by:

$$y = \sin \left ( \frac{\pi}{n}\left ( x - x_0 + \frac{n}{2}\right )\right )$$

These have period, amplitude, and phase of $2n$, $1$, and $x_0-\frac{\pi}{2}$, respectively.

Try it out here.

AndJM
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