Question

determine the remaining sides and angles of the triangle abc.
a = 117.24°, c = 31.29°, c = 130
b = 31.47°
a ≈ 222.5
(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: Find angle \( B \)

In a triangle, \( A + B + C=180^{\circ} \). So \( B = 180^{\circ}-A - C \).
Given \( A = 117.24^{\circ}\) and \( C = 31.29^{\circ}\), then \( B=180^{\circ}-117.24^{\circ}-31.29^{\circ}=31.47^{\circ}\) (already given).

Step2: Use the Law of Sines

The Law of Sines is \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\).
We know \( c = 130\), \(C = 31.29^{\circ}\), \(B = 31.47^{\circ}\).
From \(\frac{b}{\sin B}=\frac{c}{\sin C}\), we can solve for \( b\).
\(b=\frac{c\sin B}{\sin C}\)
Substitute \(c = 130\), \(B = 31.47^{\circ}\), \(C = 31.29^{\circ}\) into the formula.
\(\sin B=\sin(31.47^{\circ})\approx0.521\), \(\sin C=\sin(31.29^{\circ})\approx0.519\)
\(b=\frac{130\times\sin(31.47^{\circ})}{\sin(31.29^{\circ})}=\frac{130\times0.521}{0.519}\)
\(b=\frac{67.73}{0.519}\approx130.5\)

Answer:

\(b\approx130.5\)