Express AB in terms of $a$

Question

Hello i have these question here. Which way should i solve it? I've tried sine theorem but couldn't find it.

Given the triangle $\bigtriangleup$$ABC$ with $\angle$$BCA$$=$$\pi/3$, $\angle$$ABC$$=$$a$, $|BC|=$$\sqrt3$. Express $|AB|$ in terms of $a$.

Answer 1

You know 2 angles in the triangle so it is trivial to work out the size of $\angle BAC$ . i.e. ($\pi$ - the sum of the other 2 angles). $$\angle BAC = \pi - (\frac{\pi}{3} + a) = \frac{2\pi}{3} - a$$ You can now use the sine rule to work out the length of $|AB|$. Using the sine rule we have the following: $$\frac{\sqrt3}{sin(\frac{2\pi}{3} - a)}=\frac{|AB|}{sin(\frac{\pi}{3})}$$ Which gives us: $$sin(\frac{\pi}{3})\cdot\frac{\sqrt3}{sin(\frac{2\pi}{3} - a)}=|AB|$$ Evaluating sine, we get: $$\frac{\sqrt3}{2}\cdot\frac{\sqrt3}{sin(\frac{2\pi}{3} - a)}=|AB|$$ And finally: $$\frac{3}{2\cdot sin(\frac{2\pi}{3} - a)}=|AB|$$

Answer 2

I don't want to give a full solution so you have a chance to try it for yourself. The method to use is:$$\frac{\sin{\angle BAC}}{|BC|}=\frac{\sin{\angle BCA}}{|AB|}$$ Work out the angle $\angle BAC$ using the fact that angles in a triangle sum to $\pi$. Then this can be simplified to give you $|AB|$ in terms of some numbers and $a$.