Question

find the unknown angles in triangle abc for each triangle that exists.
a = 36.5° a = 3.2 c = 10.8
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 b = □° and c = □°
(round to the nearest tenth as needed.)
b. there are two possible sets of remaining angles. the measurements for when b is larger are b₁ = □° and c₁ = □°. the measurements for when b is smaller are b₂ =
and c₂ = □°
(round to the nearest tenth as needed.)
c. no such triangle exists.

Explanation:

Step1: Apply the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{c}{\sin C}\).
Substitute \(A = 36.5^{\circ}\), \(a = 3.2\), and \(c = 10.8\) into the formula:
\(\sin C=\frac{c\sin A}{a}=\frac{10.8\times\sin36.5^{\circ}}{3.2}\)
Calculate \(\sin36.5^{\circ}\approx0.5948\)
\(\sin C=\frac{10.8\times0.5948}{3.2}\approx\frac{6.4238}{3.2}\approx2.0074\)
Since the range of the sine function is \([- 1,1]\), and \(2.0074>1\)

Answer:

C. No such triangle exists.