Final answer:
Statement 2) a/sin(A) = b/sin(B) = c/sin(C) is the Law of Sines, not used to derive it; whereas statements 1), 3), and 4) can be rearranged to derive the Law of Sines by expressing each side with the corresponding sine of the opposite angle.
Step-by-step explanation:
The statements that are useful in deriving the Law of Sines are: 1) a² = b² + c² - 2bc * cos(A), 3) a² = b² + c² - 2bc * cos(C), and 4) a² = b² + c² - 2bc * cos(B). However, statement 2) a/sin(A) = b/sin(B) = c/sin(C) is actually the Law of Sines itself. The Law of Sines is used to find unknown sides or angles of a triangle when there is not enough information to use the Pythagorean theorem or when the triangle is not right-angled. The law states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all sides and angles in a given triangle.
For example, in any triangle ABC with sides a, b, and c and angles A, B, and C, respectively, it holds that:
- a/sin(A) = b/sin(B) = c/sin(C)
This law can be derived by using the Law of Cosines to find an expression for sin(A) and then rearrange to show the proportional relationship between sides and their opposite angles' sines.