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

• A. $\frac{a}{A}=\frac{b}{B}=\frac{c}{C}$
• B. $\frac{a}{cosA}=\frac{b}{cosB}=\frac{c}{cosC}$
• C. $\frac{a}{b}=\frac{c}{A}=\frac{B}{C}$
• D. $\frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC}$

Answer: (D)

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. For a triangle ABC with sides a, b, and c opposite angles A, B, and C respectively, the Law of Sines states that the ratio of a side length to the sine of its opposite angle is constant for all three sides.
2. Write the Law of Sines formula as: \(\frac{a}{\sin\left(A\right)} = \frac{b}{\sin\left(B\right)} = \frac{c}{\sin\left(C\right)}\).
3. Understand that this formula means the ratio of side a to \(\sin(A)\) is equal to the ratio of side b to \(\sin(B)\), and also equal to the ratio of side c to \(\sin(C)\).
4. Note that other options involving ratios with angles directly (like \(\frac{a}{A}\)) or with cosine functions (like \(\frac{a}{\cos(A)}\)) do not represent the Law of Sines and are incorrect.
5. Therefore, the correct expression of the Law of Sines is the one that uses the sine of the angles in the denominator, matching the formula \(\frac{a}{\sin\left(A\right)} = \frac{b}{\sin\left(B\right)} = \frac{c}{\sin\left(C\right)}\).