Question

which of the following statements are useful in deriving the law of sines? check all of the boxes that apply. a² = b² + h² sin a = h/c sin b = h/a sin c = h/a

Explanation:

Step1: Recall the definition of sine in a right - triangle

In a right - triangle, the sine of an angle is defined as the ratio of the length of the opposite side to the length of the hypotenuse.
For the right - triangle with angle \(A\), \(\sin A=\frac{h}{c}\) (where \(h\) is the length of the side opposite to angle \(A\) and \(c\) is the hypotenuse of the right - triangle formed).
For the right - triangle with angle \(C\), \(\sin C=\frac{h}{a}\) (where \(h\) is the length of the side opposite to angle \(C\) and \(a\) is the hypotenuse of the right - triangle formed).
From \(\sin A = \frac{h}{c}\), we can get \(h = c\sin A\). From \(\sin C=\frac{h}{a}\), we can get \(h=a\sin C\). Then \(c\sin A=a\sin C\), or \(\frac{\sin A}{a}=\frac{\sin C}{c}\).

Step2: Analyze other options

The formula \(a^{2}=b^{2}+h^{2}\) is the Pythagorean theorem. But in the context of deriving the law of sines (which is about the ratio of sines of angles and the lengths of opposite sides in a non - right triangle), this formula is not directly useful.
For \(\sin B=\frac{h}{a}\), if we consider the angle \(B\) in the wrong triangle (the right - triangle used for deriving the law of sines is not the one where this relation would hold in the standard derivation of \(\frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}\)).

Answer:

\(\sin A=\frac{h}{c}\), \(\sin C=\frac{h}{a}\)