Question

find the unknown angles in triangle abc for the following triangle if it exists.
a = 27.5° b = 42.9 ft a = 27.6 ft
select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
(round to the nearest tenth as needed.)
○ a. there are two possible sets of remaining angles. the measurements for when b is smaller are b₁ = □° and c₁ = □°. the measurements for when b is larger are b₂ = □° and c₂ = □°.
○ b. there is only one possible set of remaining angles. the measurements for the remaining angles are b = □° and c = □°
○ c. no such triangle exists

Explanation:

Step1: Apply the Law of Sines

By the Law of Sines, \(\frac{a}{\sin A}=\frac{b}{\sin B}\). Substituting the given values \(A = 27.5^{\circ}\), \(a = 27.6\) ft, and \(b = 42.9\) ft, we get \(\sin B=\frac{b\sin A}{a}\).

$$ \sin B=\frac{42.9\times\sin(27.5^{\circ})}{27.6} $$

$$ \sin B=\frac{42.9\times0.4617}{27.6}\approx\frac{19.817}{27.6}\approx0.718 $$

Step2: Find angle \(B\)

Since \(\sin B\approx0.718\), \(B=\sin^{- 1}(0.718)\approx46.0^{\circ}\) or \(B = 180^{\circ}-46.0^{\circ}=134.0^{\circ}\)

Step3: Check for valid triangles

Case 1: \(B_1 = 46.0^{\circ}\)

Using the angle - sum property of a triangle \(A + B+C=180^{\circ}\), we find \(C_1=180^{\circ}-(27.5^{\circ}+46.0^{\circ})=106.5^{\circ}\)

Case 2: \(B_2 = 134.0^{\circ}\)

Using the angle - sum property \(C_2=180^{\circ}-(27.5^{\circ}+134.0^{\circ})=18.5^{\circ}\)

Answer:

A. There are two possible sets of remaining angles. The measurements for when \(B\) is smaller are \(B_1 = 46.0^{\circ}\) and \(C_1 = 106.5^{\circ}\). The measurements for when \(B\) is larger are \(B_2 = 134.0^{\circ}\) and \(C_2 = 18.5^{\circ}\)