The Law of Cosines

Two sides and the angle between them (SAS) give no complete pair, so the law of sines is stuck. The law of cosines finds the third side, with C the angle between the sides a and b:

$${c}^{2} = {a}^{2}+{b}^{2}-2\cdot a\cdot b\cdot \cos\left(C\right)$$

It is Pythagoras with a correction. At C = 90° the cosine is 0 and the last term vanishes. An angle under 90° squeezes the sides together, the cosine is positive and the term takes some length away.

Sides 8 and 15 with 60° between them: ${c}^{2} = {8}^{2}+{15}^{2}-2\cdot 8\cdot 15\cdot \cos\left(60^{\circ}\right)$, which is 64 + 225 − 120 = 169. c² = 169 gives c = 13 or c = −13, and a length is positive, so the side is 13. On the board the negative root is ruled out on its own.