<s>[INST] Find angle in triangle $ABC$ with cevian line $AD$, such that $AB=CD$. 
As you can see in the picture, there is a triangle $ABC$ with $∠C=30°$ and $∠B=40°$. Now we assuming that $AB=CD$, try to find the exact value of $∠CAD$.

My attempt: Denote $∠CAD$ by $x$, we know that
$$\frac{\sin C}{AD}=\frac{\sin{x}}{CD},\quad\frac{\sin B}{AD}=\frac{\sin{(x+C)}}{AB}$$
Then we have ($∠C=30°=\frac{\pi}{6},∠B=40°=\frac{2\pi}{9}$)
$$\frac{\sin{\frac{\pi}{6}}}{\sin{x}}=\frac{\sin{\frac{2\pi}{9}}}{\sin{(x+\frac{\pi}{6})}}=\frac{AD}{AB}$$

Everything looks ok so far, but I have trouble solving the equation. What's more, Wolfram tells me that the answer is $x=\frac{5\pi}{18}$.
This exercise is in my sister's assignment, so I think this exercise should have a high-school (or high-school olympic) level answer.
More: The exercise appears in geometry part, so a pure geometric method will be better.
 [/INST] An algebraic proof.
Consider the triangle $ACD$ and label $\varphi=\angle\, CAD$. Both $AC$ and $AD$ can be related to $AB$, i.e., $CD$, by the law of sines: $\frac{CD}{\sin \angle CAD}=\frac{AD}{\sin \angle ACD}=2AD$, thus $AD=\frac{AB}{2\sin \varphi}$. Also, $\frac{AC}{\sin \angle ABC}=\frac{AB}{\sin \angle ACB}=2AB$, thus $AC=2\sin 40^\circ AB$. By the law of cosines,
$$CD^{2}=AC^{2}+AD^{2}-2AC\cdot AD\cos\varphi
\\
\Rightarrow
\left(4\sin^{2}40^\circ+\frac{\csc^{2}\varphi}{4}-1-{2\cot\varphi\sin40^\circ}\right)AB^{2}=0
$$
Since $AB>0$, this equation has a solution iff $\varphi$ is a root of the parenthetical term. Let $\cos\varphi=x$, $\sin40^\circ=\lambda$. Thus, equating the parenthetical term to zero gives the quadratic in $x^2$,
$$16\left(16\lambda^{4}-4\lambda^{2}+1\right)x^{4}+8\left(-64\lambda^{4}+20\lambda^{2}-3\right)x^{2}+\left(256\lambda^{4}-96\lambda^{2}+9\right)=0$$
which can be solved for
$$\begin{align}
x&=\pm\frac{1}{4}\sqrt{\frac{256\lambda ^{4}-80\lambda ^{2}+12\pm\ 8\sqrt{3}\lambda }{\left(16\lambda ^{4}-4\lambda ^{2}+1\right)}}
\\
x&\in\left\{\pm0.6428,\,\pm0.9740\right\}
\\
\varphi&\in\left\{13.12^\circ,\,50.00^\circ,\,130.00^\circ,\,166.91^\circ\right\}
\end{align}
$$
It suffices to see that $\varphi=50.00^\circ$ is the correct solution, since the angle is acute and $\varphi=13.12^\circ$ would make $\angle\,DAB=96.88^\circ$, which is absurd.
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