This image was doing the rounds on a popular text messaging application, so I decided to give it a try.
From sine rule in $\triangle ABP$: $$\frac{AB}{\sin(150^\circ)} = \frac{AP}{\sin(10^\circ} \\ \implies AP = 2AB \sin(10^\circ)$$
Applying sine rule again in $\triangle APC$: $$\frac{AP}{\sin(60^\circ + x)} = \frac{AC}{\sin(x)}$$ Manipulating the equation and using some properties gives us $$x = \arctan\left(\frac{\sqrt 3}{4\sin(10^\circ) - 1}\right)$$ This gives $x = -80^\circ$, but since it's an arctan, $x = 100^\circ$. Also, since $\sin(x) = \sin(\pi - x)$, $x = 80^\circ$ as well.
My question is: Is there a way to solve this problem that does not require a calculator? I tried to chase angles but that did not work out in this case. This solution requires computing $\sin(10^\circ)$ as well as the $\arctan$ of that expression, which needs a calculator.