Question

use the law of sines. find b to the nearest tenth. 65.1 31.3 29.7 100.2

Explanation:

Step1: Apply the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). In this triangle, we know \(a = 56\), \(A=33^{\circ}\), and \(B = 77^{\circ}\). So, \(\frac{56}{\sin33^{\circ}}=\frac{b}{\sin77^{\circ}}\).

Step2: Solve for \(b\)

First, find \(\sin33^{\circ}\approx0.5446\) and \(\sin77^{\circ}\approx0.9744\). Then, from \(\frac{56}{\sin33^{\circ}}=\frac{b}{\sin77^{\circ}}\), we can rewrite it as \(b=\frac{56\times\sin77^{\circ}}{\sin33^{\circ}}\). Substitute the values: \(b=\frac{56\times0.9744}{0.5446}\). Calculate \(56\times0.9744 = 54.5664\), then \(b=\frac{54.5664}{0.5446}\approx100.2\).

Answer:

\(100.2\) (the fourth option)