Two sides of a triangle are $5$ cm and $7$ cm and the angle between them is $60^\circ$. Find the length of the third side.

By the cosine rule $c^2 = a^2 + b^2 - 2ab\cos C = 25 + 49 - 2(5)(7)\cos 60^\circ = 74 - 35 = 39$, so $c = \sqrt{39}$ cm. Adding the last term instead of subtracting gives $\sqrt{109}$, and omitting it gives $\sqrt{74}$.