Hug de Roda

As for what @my2cts was talking about, he means that $\theta$ is a consequence of $F$, and thus, $d=0$ if and only if $F=0$. So in reality $T=0/0$.

How is this not asking about underlying physics concepts, am I missing something? How is this a closed question?

@Dennis Marx It's applying $\ln$ on both sides and then substitution $\sin x=e^z$ and you'll get something like $ze^z=\ln2$ which has the form of the inverse of Lambert function. You can do something similar with the equation OP asks but the thing is $\sin x=\sqrt{1-\cos^2x}$...

I asked the same problem on this server but with further context (like at the same time as this one) and got an answer by approximation. I'll adjunt it here: math.stackexchange.com/questions/4484480/….

Do I ask him personally or wait for him to join? What do I mention? The inversion theorem and the failed Pythoin code?

The hard part about the code is the n-1 th derivative and its limit

No success unfortunately :( : I couldn't write the code nor reduce the g_n component of the formula of the Lagrangian inverse. Furthermore, it looks like WolframAlpha can find inverses for that tan and sec equation for a values bigger than 1, but since a is between 0 and 1, then it cannot find any inverses. So non of these methods worked.

Now I should just make a Python code which can compute this long formula

For the Lagrange inverse, since it required $f'(k)≠0$, I used $k=0.05$ because when sliding the $a$ and $b$ parameters, it almost covered all y values between 0 and $\pi/2$ for the equation of $\frac{y}{a}>0$; and so, I used this small $k$ for which $a$ and $b$ are unusual, this way working for most $a$ and $b$.

Anyways I figured out on myown

What would $w$ be in $z=f(w)$ for the Lagrangian?

I'll be looking for both: the Lagrange's inverse and the one you sent me on WolframAlpha

I'll find the inverse function tomorrow morning (at least in my time zone). I'll go to sleep.

Moreover, when deriving that tan and sec equation you had to suppose $\cos{ax}≠0$, let's take that into account too.

@TymaGaidash Thats's basically the solution right? WolframAlpha gives it in form of # functions or whatever are those.

Yes I notices and thanks for the explanation I now get it.

Is this the chat you're talking about? Sorry, I am new to this.

Why would you plug in $x = 0$?

Me neither just saying. Could you elaborate on the radicals and small $a$ part? Also, I'll look into this inversion theorem.

I initially tried doing an average and even approximating, but no success. I then thought about $e^{ix}$ definition but ended up in a mess.

What about approaching it using Taylor series, Fourier series or similar?