Question

determine the remaining sides and angles of the triangle abc.
a=111.93°, c=29.47°, c=120
b= 38.6 °
a≈□
(do not round until the final answer. then round to the nearest tenth as needed.)

Explanation:

Step1: Recall the Law of Sines

The Law of Sines states that in any triangle \(ABC\), \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). We know angle \(A = 111.93^\circ\), angle \(C=29.47^\circ\), and side \(c = 120\). We need to find side \(a\). So we can use the ratio \(\frac{a}{\sin A}=\frac{c}{\sin C}\).

Step2: Solve for \(a\)

From \(\frac{a}{\sin A}=\frac{c}{\sin C}\), we can re - arrange the formula to solve for \(a\): \(a=\frac{c\times\sin A}{\sin C}\)

First, calculate \(\sin A\) and \(\sin C\):
\(\sin(111.93^\circ)=\sin(180^\circ - 68.07^\circ)=\sin(68.07^\circ)\approx0.9272\)
\(\sin(29.47^\circ)\approx0.4924\)

Then substitute \(c = 120\), \(\sin A\approx0.9272\) and \(\sin C\approx0.4924\) into the formula for \(a\):
\(a=\frac{120\times\sin(111.93^\circ)}{\sin(29.47^\circ)}=\frac{120\times0.9272}{0.4924}\)

Calculate the numerator: \(120\times0.9272 = 111.264\)

Then divide by the denominator: \(\frac{111.264}{0.4924}\approx226.0\)

Answer:

\(226.0\)