Question

law of sines: \\(\frac{\sin a}{a} = \frac{\sin b}{b} = \frac{\sin c}{c}\\)
in \\(\triangle abc\\), \\(c = 5.4\\), \\(a = 3.5\\), and \\(m\angle a = 20^\circ\\). what are the possible approximate lengths of \\(b\\)? use the law of sines to find the answer.

• 2.0 units and 4.6 units
• 2.1 units and 6.7 units
• 2.3 units and 7.8 units
• 2.6 units and 6.6 units

Explanation:

Step1: Apply the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). We know \(a = 3.3\), \(c=5.4\), and \(A = 20^{\circ}\). First, find \(\sin C\) using \(\frac{a}{\sin A}=\frac{c}{\sin C}\), so \(\sin C=\frac{c\sin A}{a}=\frac{5.4\sin20^{\circ}}{3.3}\).
\(\sin20^{\circ}\approx0.3420\), then \(\sin C=\frac{5.4\times0.3420}{3.3}=\frac{1.8468}{3.3}\approx0.5596\), so \(C\approx34^{\circ}\) or \(C = 180^{\circ}- 34^{\circ}=146^{\circ}\).

Step2: Find \(B\) for each case of \(C\)

Case 1: If \(C = 34^{\circ}\), then \(B=180^{\circ}-(A + C)=180^{\circ}-(20^{\circ}+34^{\circ})=126^{\circ}\). Using \(\frac{a}{\sin A}=\frac{b}{\sin B}\), \(b=\frac{a\sin B}{\sin A}\). \(\sin126^{\circ}=\sin(180^{\circ}- 54^{\circ})=\sin54^{\circ}\approx0.8090\), \(b=\frac{3.3\times0.8090}{0.3420}=\frac{2.6697}{0.3420}\approx7.8\).
Case 2: If \(C = 146^{\circ}\), then \(B=180^{\circ}-(A + C)=180^{\circ}-(20^{\circ}+146^{\circ})=14^{\circ}\). Using \(\frac{a}{\sin A}=\frac{b}{\sin B}\), \(\sin14^{\circ}\approx0.2419\), \(b=\frac{3.3\times0.2419}{0.3420}=\frac{0.7983}{0.3420}\approx2.3\).

Answer:

2.3 units and 7.8 units