<s>[INST] Length of a side of a triangle given the angles and the area We have $\triangle ABC$ with the following measures:
$A = 65^\circ$,
$B= 75^\circ$, 
$\text{area}= 88\,m^2$.
How can I determine the longest side?
 [/INST] We want to find $AC=b$.
We have
$$88=\frac 12ac\sin75^\circ\tag1$$
By the law of sines, we have
$$\frac{a}{\sin 65^\circ}=\frac{b}{\sin 75^\circ}=\frac{c}{\sin 40^\circ}$$
So,
$$a=\frac{\sin 65^\circ}{\sin 75^\circ}b,\quad c=\frac{\sin 40^\circ}{\sin 75^\circ}b\tag2$$
From $(1)(2)$,
$$88=\frac 12\cdot\frac{\sin 65^\circ}{\sin 75^\circ}b\cdot \frac{\sin 40^\circ}{\sin 75^\circ}b\sin75^\circ$$
Hence,
$$(\text{the longest side})=b=\sqrt{\frac{176\sin 75^\circ}{\sin 65^\circ\sin 40^\circ}}$$
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