Question

determine the remaining sides and angles of the triangle abc.
the measure of angle b is
(simplify your answer. type an integer or a decimal)
the length of side a is
(simplify your answer. type an integer or a decimal. round to the nearest tenth as needed)
the length of side b is
(simplify your answer. type an integer or a decimal. round to the nearest tenth as needed)

Explanation:

Step1: Find angle B

The sum of angles in a triangle is \(180^{\circ}\). So, \(B = 180^{\circ}-A - C\).
Given \(A = 16.8^{\circ}\) and \(C=112.5^{\circ}\), then \(B=180^{\circ}-16.8^{\circ}-112.5^{\circ}=50.7^{\circ}\).

Step2: Use the Law of Sines to find side \(a\)

The Law of Sines is \(\frac{a}{\sin A}=\frac{c}{\sin C}\). Here \(c = 59.6\) ft, \(A = 16.8^{\circ}\), \(C = 112.5^{\circ}\).
So, \(a=\frac{c\sin A}{\sin C}=\frac{59.6\times\sin(16.8^{\circ})}{\sin(112.5^{\circ})}\).
We know that \(\sin(16.8^{\circ})\approx0.287\), \(\sin(112.5^{\circ})\approx0.925\).
Then \(a=\frac{59.6\times0.287}{0.925}=\frac{17.1052}{0.925}\approx18.5\) ft.

Step3: Use the Law of Sines to find side \(b\)

By the Law of Sines \(\frac{b}{\sin B}=\frac{c}{\sin C}\).
Since \(B = 50.7^{\circ}\), \(c = 59.6\) ft, \(\sin(50.7^{\circ})\approx0.774\), \(\sin(112.5^{\circ})\approx0.925\).
Then \(b=\frac{c\sin B}{\sin C}=\frac{59.6\times0.774}{0.925}=\frac{46.1304}{0.925}\approx49.9\) ft.

Answer:

The measure of angle \(B\) is \(50.7^{\circ}\).
The length of side \(a\) is \(18.5\) ft.
The length of side \(b\) is \(49.9\) ft.