Question

determine the remaining sides and angles of the triangle abc.
a = 31.08° b = 26.18° c = 18.43 m
c = 122.74°
a ≈ 11.31 m
(round to the nearest hundredth as needed.)
b ≈ m
(round to the nearest hundredth 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 \(A = 31.08^{\circ}\), \(B=26.18^{\circ}\), \(C = 122.74^{\circ}\), and \(c = 18.43\) m. 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: \(b=\frac{18.43\times\sin(26.18^{\circ})}{\sin(122.74^{\circ})}\).
We know that \(\sin(26.18^{\circ})\approx0.442\) and \(\sin(122.74^{\circ})\approx0.845\).
Then \(b=\frac{18.43\times0.442}{0.845}\).
First, calculate \(18.43\times0.442 = 8.14606\).
Then \(b=\frac{8.14606}{0.845}\approx9.64\).

Answer:

\(b\approx9.64\) m