Given triangle ABC with sides $a$, $b$, and $c$ opposite angles $A$, $B$, and $C$ respectively, which expression represents the approximate length of side $a$ using the Law of Sines?

• A. $\frac{b\cdot sin\left(C\right)}{sin\left(A\right)}$
• B. $\frac{c\cdot sin\left(A\right)}{sin\left(B\right)}$
• C. $\frac{b\cdot sin\left(A\right)}{sin\left(B\right)}$
• D. $\frac{c\cdot sin\left(B\right)}{sin\left(C\right)}$

Answer: (C)

Verified step by step guidance

1. Recall the Law of Sines, which states that in any triangle ABC, the ratio of the length of a side to the sine of its opposite angle is constant: \(\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\).
2. To find an expression for side \(a\), start from the equality \(\frac{a}{\sin(A)} = \frac{b}{\sin(B)}\) since it relates side \(a\) and side \(b\) with their opposite angles.
3. Rearrange this equation to solve for \(a\) by multiplying both sides by \(\sin(A)\): \(a = \frac{b}{\sin(B)} \times \sin(A)\).
4. Write the expression clearly as \(a = \frac{b \cdot \sin(A)}{\sin(B)}\) to represent the length of side \(a\) in terms of side \(b\) and the sines of angles \(A\) and \(B\).
5. This formula allows you to calculate the approximate length of side \(a\) if you know the length of side \(b\) and the measures of angles \(A\) and \(B\).