Question

which expression represents the approximate length of (overline{bc})? law of sines: (\frac{sin(a)}{a}=\frac{sin(b)}{b}=\frac{sin(c)}{c}) (\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})})

Explanation:

Step1: Identify sides and angles for law of sines

In \(\triangle ABC\), let \(AB = c = 3\), \(\angle A=66^{\circ}\), \(\angle C = 38^{\circ}\), and we want to find \(BC=a\). According to the law of sines \(\frac{\sin(A)}{a}=\frac{\sin(C)}{c}\).

Step2: Cross - multiply to solve for \(a\)

Cross - multiplying the law of sines formula \(\frac{\sin(A)}{a}=\frac{\sin(C)}{c}\) gives \(a\times\sin(C)=c\times\sin(A)\). Then \(a=\frac{c\times\sin(A)}{\sin(C)}\).

Step3: Substitute the given values

Substitute \(c = 3\), \(\angle A = 66^{\circ}\), and \(\angle C=38^{\circ}\) into the formula. We get \(a=\frac{(3)\sin(66^{\circ})}{\sin(38^{\circ})}\).

Answer:

\(\frac{(3)\sin(66^{\circ})}{\sin(38^{\circ})}\) (corresponds to the first option)