How to calculate the length x when one side is not known in the cosine rule? 
the Question will become clear from this picture.
Here, the lengths of A and B are known and the angle a1 and the angle beta are also known. According to the cosine rule, ab is also known. But what is the formula for calculating the length cb when B is known and the angle beta is known?
OC is the line of construction so the length is also unknown.
 A: You know $A,B$, and $ab$. Use the sin law
$$
\frac{x}{\sin \beta}=\frac{B}{\sin \hat{Ocb}},
$$
where $\hat{Ocb}$ can be calculated again by the sin law:
$$
\left(\frac{A}{\sin (\hat{Ocb}-\beta)}=\right)\frac{A}{\sin(\pi- \hat{Ocb}-\beta)}=\frac{ab}{\sin a_1}.
$$
Therefore
\begin{align}
x=\frac{B}{\sin \hat{Ocb}}\sin \beta&=\frac{B \sin \beta}{\sin \left(\beta+\sin^{-1}\left(\frac{A\sin a_1}{ab}\right)\right)}\\
&=\frac{B \sin \beta}{\sin \left(\beta+\sin^{-1}\left(\frac{A\sin a_1}{\sqrt{A^2+B^2-2AB\cos a_1}}\right)\right)}.
\end{align}
A: Hint:
sines rule:
$$
\frac{OB}{\sin(\angle OCB)}=\frac{x}{\beta}
$$
with $\angle OCB=180°-(\beta+\angle CBO)$
A: 
\begin{align} 
2S_{\triangle AOB}&=
2S_{\triangle AOC}+2S_{\triangle COB}
,\\
a\,b\,\sin\alpha
&=
|OC|\,a\,\sin(\alpha-\beta)
+
|OC|\,b\,\sin\beta
,\\
|OC|&=
\frac{a\,b\,\sin\alpha}{a\,\sin(\alpha-\beta)+b\,\sin\beta}
,\\
x^2&=\dots
\end{align}  
A: You can write cosine rule for two different triangles $Ocb$ and $Oac$ with $Oc$ and $x$ unknown and solve them simultaneously.
