law of sines definition

1. Understand the Law of Sines: The Law of Sines is a fundamental trigonometric principle that relates the lengths of the sides of a triangle to the sines of its corresponding opposite angles. For any triangle $ABC$ with side lengths $a, b,$ and $c$ opposite to angles $A, B,$ and $C$ respectively, the law states that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides.

2. Formulate the ratios: Based on the definition, we can write the relationships as:
$$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$
Alternatively, taking the reciprocal of these ratios also maintains equality:
$$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$$

3. Compare with the given options:
- Option A: $\frac{\sin C}{a} = \frac{\sin A}{b} = \frac{\sin B}{c}$ (Incorrect)
- Option B: $\frac{\sin A}{a} = \frac{\sin C}{b} = \frac{\sin B}{c}$ (Incorrect)
- Option C: $\frac{\sin A}{b} = \frac{\sin B}{c} = \frac{\sin C}{a}$ (Incorrect)
- Option D: $\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$ (Correct)

Therefore, option D correctly represents the Law of Sines.