Given a triangle with sides $a$, $b$, $c$ opposite angles $A$, $B$, $C$ respectively, which equation can be used to find $b$ using the Law of Sines?

• A. $\frac{b}{\sin B}=\frac{c}{\sin C}$
• B. $\frac{b}{\sin B}=\frac{a}{\sin A}$
• C. $\frac{b}{\sin A}=\frac{a}{\sin B}$
• D. $\frac{b}{\sin C}=\frac{c}{\sin B}$

Answer: (B)

Verified step by step guidance

1. Recall the Law of Sines, which states that in any triangle with sides \(a\), \(b\), and \(c\) opposite angles \(A\), \(B\), and \(C\) respectively, the ratio of a side length to the sine of its opposite angle is constant. This can be written as: \[\frac{a}{\sin\left(A\right)} = \frac{b}{\sin\left(B\right)} = \frac{c}{\sin\left(C\right)}\]
2. To find an equation to solve for side \(b\), isolate the ratio involving \(b\) and its opposite angle \(B\). This gives: \[\frac{b}{\sin\left(B\right)} = \frac{a}{\sin\left(A\right)} = \frac{c}{\sin\left(C\right)}\]
3. Choose one of the other sides and angles to relate to \(b\). Typically, you use side \(a\) and angle \(A\) for convenience, so the equation becomes: \[\frac{b}{\sin\left(B\right)} = \frac{a}{\sin\left(A\right)}\]
4. This equation allows you to solve for \(b\) if you know the values of \(a\), \(A\), and \(B\). You can rearrange it to: \[b = a \times \frac{\sin\left(B\right)}{\sin\left(A\right)}\]
5. Thus, the correct Law of Sines equation to find \(b\) is: \[\frac{b}{\sin\left(B\right)} = \frac{a}{\sin\left(A\right)}\]