Which expression represents the Law of Sines in a triangle with sides a, b, c opposite angles A, B, C?

Which expression represents the Law of Sines in a triangle with sides a, b, c opposite angles A, B, C?

• A. sin A / a = sin B / b = sin C / c
• B. sin A / a = sin B / c = sin C / b
• C. a sin A = b sin B = c sin C
• D. sin A / sin B = a / b

Answer: (A)

The main idea is the Law of Sines: in any triangle, the ratio of the sine of an angle to the length of the side opposite that angle is the same for all three angles. This means the three ratios sin A / a, sin B / b, and sin C / c are equal, which is why the expression sin A / a = sin B / b = sin C / c (and its equivalent form a / sin A = b / sin B = c / sin C) correctly represents the law. If you mix up the opposite sides, such as sin A / a = sin B / c = sin C / b, the correspondence between each angle and its opposite side is broken. The form a sin A = b sin B = c sin C is not what the law states—derived from the correct relation you’d get b sin A = a sin B, not a sin A = b sin B. And while sin A / sin B = a / b holds as a consequence of the law, it only relates two pairs and doesn’t present the full, three-angle equality.

The main idea is the Law of Sines: in any triangle, the ratio of the sine of an angle to the length of the side opposite that angle is the same for all three angles. This means the three ratios sin A / a, sin B / b, and sin C / c are equal, which is why the expression sin A / a = sin B / b = sin C / c (and its equivalent form a / sin A = b / sin B = c / sin C) correctly represents the law. If you mix up the opposite sides, such as sin A / a = sin B / c = sin C / b, the correspondence between each angle and its opposite side is broken. The form a sin A = b sin B = c sin C is not what the law states—derived from the correct relation you’d get b sin A = a sin B, not a sin A = b sin B. And while sin A / sin B = a / b holds as a consequence of the law, it only relates two pairs and doesn’t present the full, three-angle equality.