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

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 $$x$$?

• I would like to thank @Batominovski for helping me take the steps above. – Elementary Dec 3 '18 at 16:11
• You can use use here compound angle formula and eleminate as far as possible the variables. – priyanka kumari Dec 3 '18 at 16:14

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

• Teacher, Are the steps correct I have taken to solve the problem? – Elementary Dec 3 '18 at 16:20
• What steps have you taken? – Robert Israel Dec 3 '18 at 16:59
• >$$\frac{\sin(x)}{\sin(\beta-x)}=\frac{\sin(\alpha)\,\sin(\theta-\gamma)}{\sin(\gamma)\,\sin(\theta-\alpha)}$$ – Elementary Dec 3 '18 at 18:20

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.

• Can you give me a link? – Elementary Dec 3 '18 at 19:24
• I'm afraid not. But if you write out your attempt at the proof, we can show you where you've gone wrong :) – Jam Dec 3 '18 at 19:31
• what is $c_1,...c_4$ – Elementary Dec 3 '18 at 19:37