Question

if $\sin(a)=h/b$, then $b\sin(a)=h$.
if $\sin(b)=h/a$, then $a\sin(b)=h$.
therefore, $b\sin(a)=a\sin(b)$.
which of the following equations is equivalent to
$b\sin(a)=a\sin(b)$?
$\frac{\sin(a)}{b}=\frac{\sin(b)}{a}$
$\frac{\sin(a)}{a}=\frac{\sin(b)}{b}$
$\frac{\sin(a)}{\sin(b)}=\frac{b}{a}$

Explanation:

Step1: Divide both sides by \(ab\)

Given \(b\sin(A)=a\sin(B)\), divide each term by \(ab\).

$$ \frac{b\sin(A)}{ab}=\frac{a\sin(B)}{ab} $$

Step2: Simplify the fractions

Simplify \(\frac{b\sin(A)}{ab}\) to \(\frac{\sin(A)}{a}\) and \(\frac{a\sin(B)}{ab}\) to \(\frac{\sin(B)}{b}\).

$$ \frac{\sin(A)}{a}=\frac{\sin(B)}{b} $$

Answer:

\(\frac{\sin(A)}{a}=\frac{\sin(B)}{b}\) (the second option)