Question

find the unknown angles in triangle abc for the following triangle if it exists.
a = 25.5° b = 44.9 ft a = 25.3 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.)
o a. there is only one possible set of remaining angles. the measurements for the remaining angles are b = and c =
o b. there are two possible sets of remaining angles. the measurements for when b is smaller are b, = and c, = the measurements for when b is larger are b, =
and c, =
o c. no such triangle exists.

Explanation:

Step1: Apply the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}\).
Substitute \(A = 25.5^{\circ}\), \(a = 25.3\) ft, and \(b = 44.9\) ft into the formula: \(\sin B=\frac{b\sin A}{a}\).

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Step2: Find the value of \(B\)

Since \(\sin B\approx0.764\), \(B = \sin^{- 1}(0.764)\approx49.8^{\circ}\) or \(B = 180^{\circ}-49.8^{\circ}=130.2^{\circ}\)

Step3: Check for the existence of triangles

For \(B_1 = 49.8^{\circ}\):
\(C_1=180^{\circ}-A - B_1=180^{\circ}-25.5^{\circ}-49.8^{\circ}=104.7^{\circ}\)
For \(B_2 = 130.2^{\circ}\):
\(C_2=180^{\circ}-A - B_2=180^{\circ}-25.5^{\circ}-130.2^{\circ}=24.3^{\circ}\)

Answer:

B. There are two possible sets of remaining angles. The measurements for when \(B\) is smaller are \(B_1 = 49.8^{\circ}\) and \(C_1 = 104.7^{\circ}\). The measurements for when \(B\) is larger are \(B_2 = 130.2^{\circ}\) and \(C_2 = 24.3^{\circ}\)