Question

find the unknown angles in triangle abc for each triangle that exists.
b = 142.2°, c = 8.1, b = 14.4
select the correct choice below, and, if necessary, fill in the answer boxes to complete your choice.
oa. there are two possible sets of remaining angles. the measurements for when a is larger are a₁ = ° and c₁ = °. the measurements for when a is smaller are a₂ = °
and c₂ = °
(round to the nearest tenth as needed.)
ob. there is only one possible set of remaining angles. the measurements for the remaining angles are a = ° and c = °
(round to the nearest tenth as needed.)
oc. no such triangle exists.

Explanation:

Step1: Apply the Law of Sines

The Law of Sines states that \(\frac{\sin B}{b}=\frac{\sin C}{c}\).
Substitute \(B = 142.2^{\circ}\), \(b = 14.4\), and \(c = 8.1\) into the formula:
\(\sin C=\frac{c\sin B}{b}=\frac{8.1\times\sin(142.2^{\circ})}{14.4}\)
First, calculate \(\sin(142.2^{\circ})=\sin(180^{\circ}- 37.8^{\circ})=\sin(37.8^{\circ})\approx0.613\)
Then \(\sin C=\frac{8.1\times0.613}{14.4}\approx\frac{4.9653}{14.4}\approx0.345\)
So \(C=\sin^{-1}(0.345)\approx20.2^{\circ}\)

Step2: Check for the ambiguous case

Since \(B = 142.2^{\circ}\) (an obtuse angle) and \(b>c\) (because \(14.4>8.1\)), there is only one - triangle.

Step3: Calculate angle \(A\)

We know that \(A + B + C=180^{\circ}\), so \(A=180^{\circ}-B - C\)
Substitute \(B = 142.2^{\circ}\) and \(C = 20.2^{\circ}\)
\(A=180^{\circ}-142.2^{\circ}-20.2^{\circ}=17.6^{\circ}\)

Answer:

B. There is only one possible set of remaining angles. The measurements for the remaining angles are \(A = 17.6^{\circ}\) and \(C=20.2^{\circ}\)