Question

determine the remaining sides and angles of the triangle abc.
a = 101.71°, c = 24.01°, c = 160
b = 54.28°
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 \(\frac{a}{\sin A}=\frac{c}{\sin C}\).
We know \(A = 101.71^{\circ}\), \(C=24.01^{\circ}\), and \(c = 160\).
So, \(a=\frac{c\sin A}{\sin C}\).

Step2: Substitute the values

Substitute \(c = 160\), \(\sin A=\sin(101.71^{\circ})\), and \(\sin C=\sin(24.01^{\circ})\) into the formula.
\(\sin(101.71^{\circ})\approx0.976\), \(\sin(24.01^{\circ})\approx0.407\).
\(a=\frac{160\times0.976}{0.407}\).

Step3: Calculate the value of \(a\)

First, calculate \(160\times0.976 = 156.16\).
Then, \(a=\frac{156.16}{0.407}\approx383.7\).

Answer:

\(a\approx383.7\)