In any triangle, which relationship must be true according to the $\mathrm{Law} \mathrm{of} \mathrm{Sines}$?

• A. $\sin A+\sin B+\sin C=1$
• B. $\frac{a}{b}=\frac{\sin A}{\sin B}$
• C. $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$
• D. $a+b+c=180$

Answer: (C)

Verified step by step guidance

1. Recall that the Law of Sines relates the sides and angles of any triangle by stating that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides. This means we are looking for an equation involving \( a, b, c \) and \( \sin A, \sin B, \sin C \) in a proportional relationship.
2. Write down the Law of Sines formula: \[\frac{a}{\sin\ A} = \frac{b}{\sin\ B} = \frac{c}{\sin\ C}\] This shows that each side length divided by the sine of its opposite angle is equal to the same constant.
3. Understand why the other options are incorrect: - The sum of sines of angles \( \sin A + \sin B + \sin C = 1 \) is not generally true for triangles. - The ratio \( \frac{a}{b} = \frac{\sin A}{\sin B} \) is only part of the Law of Sines but incomplete without including side \( c \) and angle \( C \). - The sum of sides \( a + b + c = 180 \) is incorrect; it is the sum of angles that equals 180 degrees, not the sides.
4. Recognize that the Law of Sines is useful for solving triangles when you know either two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA). It helps find unknown sides or angles by setting up proportions based on the formula.
5. Summarize that the key relationship according to the Law of Sines is: \[\frac{a}{\sin\ A} = \frac{b}{\sin\ B} = \frac{c}{\sin\ C}\] which must hold true for any triangle.