Question

find the unknown angles in triangle abc for the following triangle if it exists
c = 47° 30, b = 22.7 m, c = 38.1 m
select the correct choice below, and, if necessary, fill in the answer boxes to complete your choice.
a. there is only one possible set of remaining angles. the measurements for the remaining angles are a = □°□, and b = □°□.
(do not round until the final answers. then round to the nearest whole number as needed.)
b. there are two possible sets of remaining angles. the measurements for when b is larger are a₁ = □°□, and b₁ = □°□. the measurements for when b is smaller are a₂ = □°□, and b₂ = □°□.
(do not round until the final answers. then round to the nearest whole number as needed.)
c. no such triangle exists.

Explanation:

Step1: Convert angle C to decimal

We know that \(C = 47^{\circ}30'\). Since \(1^{\circ}=60'\), then \(30'=\frac{30}{60}=0.5^{\circ}\), so \(C = 47.5^{\circ}\)

Step2: Use the Law of Sines

The Law of Sines is \(\frac{\sin B}{b}=\frac{\sin C}{c}\). Substituting \(b = 22.7\), \(c = 38.1\), and \(C=47.5^{\circ}\) into the formula, we get \(\sin B=\frac{b\sin C}{c}=\frac{22.7\times\sin(47.5^{\circ})}{38.1}\)

First, calculate \(\sin(47.5^{\circ})\approx0.737\)

Then \(\sin B=\frac{22.7\times0.737}{38.1}=\frac{16.7299}{38.1}\approx0.439\)

So \(B=\sin^{- 1}(0.439)\approx26.0^{\circ}\) or \(B = 180^{\circ}-26.0^{\circ}=154.0^{\circ}\)

But if \(B = 154.0^{\circ}\), then \(A=180^{\circ}-(154.0^{\circ}+47.5^{\circ})=180^{\circ}-201.5^{\circ}\), which is not possible. So there is only one - triangle.

Step3: Calculate angle A

Since \(A + B + C=180^{\circ}\), and \(B\approx26^{\circ}\), \(C = 47.5^{\circ}\)

\(A=180^{\circ}-(26^{\circ}+47.5^{\circ})=106.5^{\circ}\)

Answer:

A. There is only one possible set of remaining angles. The measurements for the remaining angles are \(A = 107^{\circ}\), \(B = 26^{\circ}\)