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Solving a trigonometric equation with two variables in Mathematica

Posted 1 year ago
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I try to solve the following equation for x since yesterday but apparently my math skills are no where near good enought to understand what's going wrong here.

I started with the following Equation in Wolfram|Alpha:

After that I tried to solve: Solve[2 (x + pi (-1 + y)) == Sin[2 x]] in Mathematica, but it just tells me that I can't solve it and I should use Reduce, but that ended in the same problem.

Adding some additional information in the form of 0 <= x <= pi && 0 <= y <= 1 didn't help either.

After some time on Google I found that it's probably because it's a transcendental function and that's where I'm stuck now. Is there even a solution to this?

5 Replies

Hi Patrick, Here is the solution, identical for both alternatives. You need to solve for both x & y, or only y.

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Most transcendental equation can't be solved analytically, we use NSolve :

f[y_] := NSolve[-2 \[Pi] + 2 x + 2 \[Pi] y == Sin[2 x] && 0 <= x <= Pi, x]
f[1/2] (*For y=1/2 we have: *)
{{x -> 1.5708}}(* Pi/2*)

And Plot for range: 0<=y<=1

Plot[{y /. f[y], y}, {y, 0, 1}, PlotStyle -> {Red, {Dashed, Black}}](*Just executed this*)

Hi Patrick, Here is the solution, identical for both alternatives. You need to solve for both x & y, or only y.

Hi Christos, thanks for the quick reply. Why do i need to solve for y? y is a paremeter i want to be able to enter to get the coresponding x. Basically the 1380 is the max. output power of my system and y represents the percentage of the max. power that i want to use (hence why y is 0 <= y <= 1). My plan was to get x = f(y) in a simple enought way to run the calculation on a microcontroller. But as it looks, i probably have to pre calculate every x for ever y between 0 and 1 in 0.01 steps and use a lookup table. Because i don't think i can run these calculations in a reasonable time on a microcontroller.

And Plot for range: 0<=y<=1

Thanks Mariusz, i used that to output a list of all x values i need and made a lookup table out of them.

As Mariusz formulated, you need to specify the value of parameter y, in order to get your x. But you can't invert that function. For instance with y = 0.5, you get x -> 1.5708, but if you set x = 1.5708, you get y = 0.5 and y = 1.5, as the two pair of solutions show.

Ah ok, makes sense.

Edit: If anyone is interested, here is the notebook i used to export the 101 values i'm interested in to a file to create the lookup table:

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