Question

the law of sines
two right triangles, △abd and △acd, were created from △abc by constructing (overline{ad} perp overline{cb}).
follow these steps to prove that (\frac{sin(b)}{b} = \frac{sin(c)}{c}).

1. for △abd, (sin(b) =) dropdown
2. multiply both sides of the equation by (c) and simplify. the equation (sin(b) = \frac{h}{c}) becomes text
3. for △acd, (sin(c) =) dropdown with options b/h, h/b

image of triangle with vertices a, b, c, d (right angle at d), height (h) (ad), side (b) (ac), side (c) (ab)

Explanation:

Step1: Recall sine in right triangle

In right triangle \( \triangle ACD \), \( \sin(C) \) is defined as the ratio of the opposite side to the hypotenuse. The opposite side to angle \( C \) is \( h \), and the hypotenuse is \( b \). So by the definition of sine in a right triangle (\( \sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}} \)), we have \( \sin(C)=\frac{h}{b} \).

Answer:

\(\frac{h}{b}\)