Question

use the law of sines to find the length of side b. round your answer to the nearest tenth.

a. 24.5
b. 36.6
c. 54.9
d. 96.5

Explanation:

Step1: Find angle \( B \)

The sum of angles in a triangle is \( 180^{\circ} \). So \( B=180^{\circ}-(42^{\circ} + 41.5^{\circ})=96.5^{\circ} \).

Step2: Apply the Law of Sines

The Law of Sines states \( \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C} \). Here, we have \( \frac{b}{\sin B}=\frac{37}{\sin A} \). Substituting \( A = 42^{\circ}\), \(B = 96.5^{\circ}\) and the side opposite \(A\) (assuming side \(b\) is opposite \(B\) and the given side \(37\) is opposite \(A\) - based on the Law of Sines formula application for the problem). So \( b=\frac{37\times\sin(96.5^{\circ})}{\sin(42^{\circ})}\).
Calculate \(\sin(96.5^{\circ})\approx0.993\) and \(\sin(42^{\circ})\approx0.669\). Then \(b=\frac{37\times0.993}{0.669}=\frac{36.741}{0.669}\approx54.9\).

Answer:

C. 54.9