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: \(\sin C=\frac{c\sin A}{a}\).
So, \(\sin C=\frac{10.8\times\sin(36.5^{\circ})}{3.2}\).
Calculate \(\sin(36.5^{\circ})\approx0.5948\).
Then \(\sin C=\frac{10.8\times0.5948}{3.2}=\frac{6.42384}{3.2}\approx2.00745\).
Since the range of the sine function is \([- 1,1]\), and \(2.00745>1\).

Answer:

C. No such triangle exists.