Is it possible to find a closed form for $x$?

Question

To solve the problem, I followed the following steps:

Is it possible to find a closed form for $x$?

$$\frac{\sin(x)}{\sin(\beta-x)}=\frac{\sin(\alpha)\,\sin(\theta-\gamma)}{\sin(\gamma)\,\sin(\theta-\alpha)}$$

where, $$x:= \angle OBC,\beta:=\angle ABC ,\alpha:=\angle OAC, \gamma:=\angle OCA ,\theta:=\angle BAC=\angle ACB$$

Mathematica says that,

$x\approx 0.033921 \approx 1.94353^\circ\,$

Is there another method to find $x$?

Answer 1

If everything except $x$ is known, $$ \frac{\sin(x)}{\sin(\beta-x)}=c$$ reduces to $$ \tan(x) = \frac{c \sin(\beta)}{1+ c \cos(\beta)}$$

Answer 2

Given that you haven't shown your steps, we can't know whether you've used a logical method but I think you've found the angle incorrectly. Nevertheless, a way you could find it could be:

1. Find. $\angle BAC$, $\angle BCA$ using the fact that the triangle is isosceles
2. Find $\angle BAO$ and $\angle BCO$
3. Find $\angle AOB$ and $\angle COB$
4. Relate $\angle BAO$, $\angle AOB$, $\angle BCO$ and $\angle AOC$ with each other using the sine rule.

You can get a solution of the form $\frac{\sin(x+c_1)}{\sin(x+c_2)}=\frac{\sin(c_3)}{\sin(c_4)}$, where $c_{1-4}$ are constants. However this equation doesn't seem to easily manipulate into a simple solution for $x$. I inputted the solution into WolframAlpha and it finds a closed form, albeit a nasty expression with $9$ terms.