To determine the statements that are useful in deriving the law of sines, let's review each statement one by one:
1. \(a^2 = b^2 + h^2\)
2. \(\sin A = \frac{h}{c}\)
3. \(\sin B = \frac{h}{a}\)
4. \(\operatorname{sin} C = \frac{h}{a}\)
The law of sines states that in any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. Mathematically, it is given by:
[tex]\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\][/tex]
Now let’s go through each statement to check its relevance:
1. \(a^2 = b^2 + h^2\):
This equation represents the Pythagorean theorem and pertains to the relationship between the sides of a right triangle. However, it's not directly useful in deriving the law of sines, which deals with ratios involving sine functions and triangle sides. Therefore, this statement is not useful.
2. \(\sin A = \frac{h}{c}\):
This statement describes the sine of angle \(A\) in a right triangle, where \(h\) is the height (or altitude) corresponding to the base \(c\). This is useful for deriving the sine ratios in the law of sines. Therefore, this statement is useful.
3. \(\sin B = \frac{h}{a}\):
Similarly, this statement describes the sine of angle \(B\) where \(h\) is the height (or altitude) corresponding to the base \(a\). This too helps in establishing the sine ratios in the law of sines. Therefore, this statement is useful.
4. \(\operatorname{sin} C = \frac{h}{a}\):
This statement describes the sine of angle \(C\) involving sides \(h\) and \(a\). However, there seems to be an inconsistency because \(C\) should ideally involve sides related to the base and height differently. In an ideal form, for consistency with other trigonometric definitions, it would be irrelevant or possibly incorrect in its current form.
Based on the correct or directly relevant trigonometric definitions involving sine and side lengths of the triangle, the useful statements are:
- \(\sin A = \frac{h}{c}\)
- [tex]\(\sin B = \frac{h}{a}\)[/tex]
