Question

determine the remaining sides and angles of the triangle abc. a = 101.71°, c = 24.01°, c = 160 b = 54.28° a ≈ 385.0 (do not round until the final answer. then round to the nearest tenth as needed.) b ≈ (do not round until the final answer. then round to the nearest tenth as needed.)

Explanation:

Step1: Use the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). We know \(A = 101.71^{\circ}\), \(C=24.01^{\circ}\), \(c = 160\), and \(B = 54.28^{\circ}\). We want to find \(b\), so we use \(\frac{b}{\sin B}=\frac{c}{\sin C}\).

Step2: Solve for \(b\)

Rearrange the formula to \(b=\frac{c\sin B}{\sin C}\). Substitute the values: \(b=\frac{160\times\sin(54.28^{\circ})}{\sin(24.01^{\circ})}\).
First, calculate \(\sin(54.28^{\circ})\approx0.811\) and \(\sin(24.01^{\circ})\approx0.407\).
Then \(b=\frac{160\times0.811}{0.407}=\frac{129.76}{0.407}\approx319.066\approx319.1\)

Answer:

\(b\approx319.1\)