Question

determine the remaining sides and angles of the triangle abc.
a = 100.45°, c = 28.94°, c = 230
b = 50.61°
a ≈ 467.4
(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 \(A = 100.45^{\circ}\), \(C=28.94^{\circ}\), \(c = 230\), and \(B=50.61^{\circ}\) (since \(A + B + C=180^{\circ}\), so \(B=180-(100.45 + 28.94)=50.61^{\circ}\)).
We want to find \(b\), so we use \(\frac{b}{\sin B}=\frac{c}{\sin C}\).

Step2: Solve for \(b\)

From \(\frac{b}{\sin B}=\frac{c}{\sin C}\), we can express \(b\) as \(b=\frac{c\sin B}{\sin C}\).
Substitute \(c = 230\), \(B = 50.61^{\circ}\), \(C=28.94^{\circ}\) into the formula.
\(\sin B=\sin(50.61^{\circ})\approx0.772\), \(\sin C=\sin(28.94^{\circ})\approx0.484\).
\(b=\frac{230\times\sin(50.61^{\circ})}{\sin(28.94^{\circ})}=\frac{230\times0.772}{0.484}\).
First, calculate \(230\times0.772 = 177.56\).
Then, \(\frac{177.56}{0.484}\approx367.0\).

Answer:

\(b\approx367.0\)