Question

ivan began to prove the law of sines using the diagram and equations below. which equation is equivalent to the equation ( bsin(a)=asin(b) )? ( \frac{a}{sin(b)}=\frac{b}{sin(a)} ) ( \frac{sin(a)}{a}=\frac{sin(b)}{b} ) ( \frac{sin(a)}{sin(b)}=\frac{b}{a} ) ( \frac{sin(b)}{a}=\frac{sin(a)}{b} ) ( sin(a)=h / b ), so ( bsin(a)=h ). ( sin(b)=h / a ), so ( asin(b)=h ). therefore, ( bsin(a)=asin(b) ).

Explanation:

Step1: Divide both sides by $\sin(A)\sin(B)$

Given \(b\sin(A)=a\sin(B)\), divide both sides by \(\sin(A)\sin(B)\).

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

Step2: Simplify the equation

Simplify the left - hand side \(\frac{b\sin(A)}{\sin(A)\sin(B)}=\frac{b}{\sin(B)}\), and the right - hand side \(\frac{a\sin(B)}{\sin(A)\sin(B)}=\frac{a}{\sin(A)}\).

Answer:

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