Question

determine the remaining sides and angles of the triangle abc.
a = 117.24°, c = 31.29°, c = 130
b = 31.47°
a ≈
(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 \(\frac{a}{\sin A}=\frac{c}{\sin C}\).

Step2: Substitute the known values

We know \(A = 117.24^{\circ}\), \(C=31.29^{\circ}\), and \(c = 130\). So \(a=\frac{c\sin A}{\sin C}\).

$$ a=\frac{130\times\sin(117.24^{\circ})}{\sin(31.29^{\circ})} $$

Step3: Calculate \(\sin(117.24^{\circ})\) and \(\sin(31.29^{\circ})\)

\(\sin(117.24^{\circ})\approx0.89\), \(\sin(31.29^{\circ})\approx0.52\)

$$ a=\frac{130\times0.89}{0.52} $$

Step4: Perform the calculation

$$ a=\frac{115.7}{0.52}\approx222.5 $$

Answer:

\(a\approx222.5\)