Which of the following triangles can you use the Law of Cosines to solve for a missing side?

• A. A triangle with only two angles and one side known
• B. A triangle with sides $a$, $b$, and included angle $C$ known
• C. A right triangle with only the two legs known
• D. A triangle with all three angles known but no sides

Answer: (B)

Verified step by step guidance

1. Recall that the Law of Cosines is used to find a missing side or angle in a triangle when you know either two sides and the included angle or all three sides.
2. Identify the known elements in each triangle option: for the Law of Cosines, you need either two sides and the included angle or three sides.
3. For the triangle with two angles and one side known, note that the Law of Cosines is not directly applicable because you don't have two sides and the included angle.
4. For the triangle with sides \(a\), \(b\), and included angle \(C\) known, this fits the Law of Cosines criteria perfectly, since you have two sides and the included angle.
5. For the right triangle with two legs known, you can use the Pythagorean theorem instead of the Law of Cosines, and for the triangle with all three angles known but no sides, you cannot use the Law of Cosines because no side lengths are given.