Which of the following statements is true about using the $\mathrm{Law} \mathrm{of} \mathrm{Sines}$ to solve a triangle when given all three side lengths?

• A. The $\mathrm{Law} \mathrm{of} \mathrm{Sines}$ can always be used directly to solve a triangle when given three side lengths.
• B. The $\mathrm{Law} \mathrm{of} \mathrm{Sines}$ is used only for right triangles with three side lengths known.
• C. The $\mathrm{Law} \mathrm{of} \mathrm{Sines}$ cannot be directly used to solve a triangle when only the three side lengths are given.
• D. The $\mathrm{Law} \mathrm{of} \mathrm{Sines}$ is only used when all three angles are given.

Answer: (C)

Verified step by step guidance

1. Recall that the Law of Sines states: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\), where \(a\), \(b\), and \(c\) are the sides opposite angles \(A\), \(B\), and \(C\) respectively.
2. Understand that to use the Law of Sines effectively, you need at least one angle-side pair (an angle and its opposite side) to relate the sides and angles.
3. When only the three side lengths are given (SSS case), no angle-side pair is directly known, so the Law of Sines cannot be applied immediately without first finding an angle by another method.
4. In the SSS case, the Law of Cosines is typically used first to find one angle, because it relates all three sides directly to an angle: \(\cos A = \frac{b^2 + c^2 - a^2}{2bc}\).
5. Once one angle is found using the Law of Cosines, the Law of Sines can then be used to find the remaining angles and sides if needed.