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Ivan began to prove the law of sines using the diagram and equations below.

[tex]\[
\begin{array}{l}
\sin (A) = \frac{h}{b} \quad \Rightarrow \quad b \sin (A) = h \\
\sin (B) = \frac{h}{a} \quad \Rightarrow \quad a \sin (B) = h
\end{array}
\][/tex]

Therefore, [tex]\(b \sin (A) = a \sin (B)\)[/tex].

Which equation is equivalent to [tex]\(b \sin (A) = a \sin (B)\)[/tex]?

A. [tex]\(\frac{a}{\sin (B)} = \frac{b}{\sin (A)}\)[/tex]

B. [tex]\(\frac{\sin (A)}{a} = \frac{\sin (B)}{b}\)[/tex]

C. [tex]\(\frac{\sin (A)}{\sin (B)} = \frac{b}{a}\)[/tex]

D. [tex]\(\frac{\sin (B)}{a} = \frac{\sin (A)}{b}\)[/tex]



Answer :

GinnyAnswer
To determine an equivalent equation to [tex]\( b \sin(A) = a \sin(B) \)[/tex], let's go through a detailed, step-by-step reasoning process:

1. Starting Equation:
[tex]\[ b \sin(A) = a \sin(B) \][/tex]

2. To manipulate this equation into a different form, let's divide both sides by [tex]\(\sin(A) \sin(B)\)[/tex]:
[tex]\[ \frac{b \sin(A)}{\sin(A) \sin(B)} = \frac{a \sin(B)}{\sin(A) \sin(B)} \][/tex]

3. Simplifying both sides:
[tex]\[ \frac{b}{\sin(B)} = \frac{a}{\sin(A)} \][/tex]

This is the resulting equation that is equivalent to the original equation [tex]\( b \sin(A) = a \sin(B) \)[/tex].

Thus, the correct answer is:
[tex]\[ \frac{a}{\sin(B)} = \frac{b}{\sin(A)} \][/tex]

By following this reasoning step-by-step, we confirm that the equivalent equation is indeed:
[tex]\[ \frac{a}{\sin(B)} = \frac{b}{\sin(A)} \][/tex]

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