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 $\frac{\sin(b)}{b} = \frac{\sin(c)}{c}$.
equation $\sin(c) = \frac{h}{b}$ becomes $b \sin(c) = h$ (partial text).

1. by the substitution property, we can set the expressions equal to each other: $c \sin(b) = b \sin(c)$
2. divide both sides of the equation by dropdown with options b, c, bc and simplify to get $\frac{\sin(b)}{b} = \frac{\sin(c)}{c}$

diagram: triangle abc with ad ⊥ cb, d is foot, h=ad, b=ac, c=ab, c, d, b on base

Explanation:

Step1: Analyze the equation

We have the equation \( c\sin(B)=b\sin(C) \) and we want to transform it into \( \frac{\sin(B)}{b}=\frac{\sin(C)}{c} \).

Step2: Determine the divisor

To get from \( c\sin(B) = b\sin(C) \) to \( \frac{\sin(B)}{b}=\frac{\sin(C)}{c} \), we need to divide both sides of the equation \( c\sin(B)=b\sin(C) \) by \( bc \). Let's check:
If we divide left - hand side \( c\sin(B) \) by \( bc \), we get \( \frac{c\sin(B)}{bc}=\frac{\sin(B)}{b} \).
If we divide right - hand side \( b\sin(C) \) by \( bc \), we get \( \frac{b\sin(C)}{bc}=\frac{\sin(C)}{c} \).

Answer:

\( bc \)