Question

find the unknown angles in triangle abc for the following triangle if it exists.
a = 25.3° b = 43.4 ft a = 27.9 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 b₁ = ° and c₁ = °. the measurements for when b is larger are b₂ = ° and c₂ = °
o c. no such triangle exists.

Explanation:

Step1: Apply the Law of Sines

By the Law of Sines, $\frac{\sin A}{a}=\frac{\sin B}{b}$. Substitute $A = 25.3^{\circ}$, $a = 27.9$ ft, and $b = 43.4$ ft:
$$\sin B=\frac{b\sin A}{a}=\frac{43.4\times\sin(25.3^{\circ})}{27.9}$$
Calculate $\sin(25.3^{\circ})\approx0.427$:
$$\sin B=\frac{43.4\times0.427}{27.9}\approx\frac{18.5318}{27.9}\approx0.664$$

Step2: Find angle B

Since $\sin B\approx0.664$, $B=\sin^{-1}(0.664)\approx41.6^{\circ}$ or $B = 180^{\circ}- 41.6^{\circ}=138.4^{\circ}$

Step3: Check for valid triangles

For $B_1 = 41.6^{\circ}$:
$C_1=180^{\circ}-A - B_1=180^{\circ}-25.3^{\circ}-41.6^{\circ}=113.1^{\circ}$
For $B_2 = 138.4^{\circ}$:
$C_2=180^{\circ}-A - B_2=180^{\circ}-25.3^{\circ}-138.4^{\circ}=16.3^{\circ}$

Answer:

B. There are two possible sets of remaining angles. The measurements for when B is smaller are $B_{1} = 41.6^{\circ}$ and $C_{1}=113.1^{\circ}$. The measurements for when B is larger are $B_{2}=138.4^{\circ}$ and $C_{2}=16.3^{\circ}$