Question

which expression represents the approximate length of $overline{bc}$?○ $\frac{(3)sin(66^{circ})}{sin(38^{circ})}$○ $\frac{sin(66^{circ})}{(3)sin(38^{circ})}$○ $\frac{(3)sin(38^{circ})}{sin(66^{circ})}$○ $\frac{sin(38^{circ})}{(3)sin(66^{circ})}$law of sines: $\frac{sin(a)}{a}=\frac{sin(b)}{b}=\frac{sin(c)}{c}$

Explanation:

Step1: Match sides to angles

In $\triangle ABC$, side $\overline{BC}$ is opposite $\angle A$ ($66^\circ$), and side $\overline{AB}=3$ is opposite $\angle C$ ($38^\circ$).

Step2: Apply Law of Sines

Set up the proportion:
$$\frac{\sin(A)}{\overline{BC}} = \frac{\sin(C)}{\overline{AB}}$$
Substitute known values:
$$\frac{\sin(66^\circ)}{\overline{BC}} = \frac{\sin(38^\circ)}{3}$$

Step3: Solve for $\overline{BC}$

Rearrange the equation to isolate $\overline{BC}$:
$$\overline{BC} = \frac{(3)\sin(66^\circ)}{\sin(38^\circ)}$$

Answer:

$\frac{(3)\sin(66^\circ)}{\sin(38^\circ)}$ (Option A)