Question

2025 geometry b wwva
w of sines
which expression represents the length of (overline{bc})?
diagram: triangle abc with vertex a at bottom left (66°), vertex c at bottom right (38°), vertex b at top; side ab labeled 3
law of sines: (\frac{sin(a)}{a} = \frac{sin(b)}{b} = \frac{sin(c)}{c}) (approximate, original may have typo)
options: (\frac{(3)sin(66^circ)}{sin(38^circ)}), (\frac{sin(38^circ)}{(3)sin(66^circ)}), (\frac{sin(66^circ)}{(3)sin(38^circ)}) (and other unshown options)

Explanation:

Step1: Identify sides and angles

In triangle \(ABC\), side \(AB = 3\), \(\angle A=66^\circ\), \(\angle C = 38^\circ\). We need to find \(BC\) (let's call it \(a\) corresponding to \(\angle A\), wait no: in Law of Sines, \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\), where \(a\) is opposite \(\angle A\), \(b\) opposite \(\angle B\), \(c\) opposite \(\angle C\). Wait, \(BC\) is opposite \(\angle A\)? Wait, no: vertex \(A\), \(B\), \(C\). So side \(BC\) is opposite \(\angle A\) (since \(BC\) is between \(B\) and \(C\), so opposite angle \(A\)), side \(AB\) is opposite \(\angle C\) (since \(AB\) is between \(A\) and \(B\), opposite angle \(C\)). So \(AB = c = 3\), \(\angle C=38^\circ\), \(\angle A = 66^\circ\), and \(BC = a\) (opposite \(\angle A\)).

Step2: Apply Law of Sines

Law of Sines: \(\frac{a}{\sin A}=\frac{c}{\sin C}\). We need to solve for \(a\) (which is \(BC\)). So rearrange: \(a=\frac{c\sin A}{\sin C}\). Substitute \(c = 3\), \(\angle A = 66^\circ\), \(\angle C=38^\circ\). So \(BC=\frac{3\sin(66^\circ)}{\sin(38^\circ)}\).

Answer:

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