Given triangle $ABC$ with sides $a$, $b$, $c$ opposite angles $A$, $B$, $C$ respectively, which of the following correctly expresses the Law of Sines?

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

Answer: (B)

Verified step by step guidance

1. Recall that the Law of Sines relates the sides of a triangle to the sines of their opposite angles. It states that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides and angles in the triangle.
2. Write down the Law of Sines formula: \(\frac{a}{\sin\!A} = \frac{b}{\sin\!B} = \frac{c}{\sin\!C}\), where \(a\), \(b\), and \(c\) are the sides opposite angles \(A\), \(B\), and \(C\) respectively.
3. Understand that this means each side divided by the sine of its opposite angle gives the same value, which can be used to find unknown sides or angles in a triangle when some measurements are known.
4. Compare the given options to this formula carefully. The correct expression must have the sine function applied to the angles in the denominator, paired with their opposite sides in the numerator.
5. Note that expressions like \(\frac{a}{b} = \frac{c}{A}\) or \(\frac{a}{A} = \frac{b}{B}\) are incorrect because they do not involve the sine of the angles, which is essential in the Law of Sines.