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For any oblique triangle, which may be acute or obtuse, if I know length of two sides, and I know all three angles, how can I determine length of unknown side?

All three sides may or may not be equal length.

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Using Law of Sines:

missing length c = known length b X sin C / sin B where A,B,C and triangle's angles, and b is angle opposite B.

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You may use the law of sines or the law of cosines. If $ABC$ is a triangle and the only unknown length is the one of $c=AB$, from $$ \frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}=\frac{1}{2R} \tag{1}$$ we get $c= b\,\frac{\sin C}{\sin B} = a\,\frac{\sin C}{\sin A}=2R\,\sin C$. We also have: $$ c^2 = a^2+b^2-2ab\cos C.\tag{2}$$

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