To solve the triangle given that [tex]\(\angle A = 109^\circ\)[/tex], [tex]\(a = 28\)[/tex], and [tex]\(b = 18\)[/tex], we will use the Law of Sines, which states:
[tex]\[
\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}
\][/tex]
First, we need to find [tex]\(\angle B\)[/tex]:
1. Step 1: Use the Law of Sines to find [tex]\(\sin B\)[/tex]:
[tex]\[
\sin B = \frac{b \cdot \sin A}{a}
\][/tex]
Given:
[tex]\[
\sin A = \sin(109^\circ)
\][/tex]
Calculate [tex]\(\sin B\)[/tex]:
[tex]\[
\sin B = \frac{18 \cdot \sin(109^\circ)}{28}
\][/tex]
2. Step 2: Calculate [tex]\(B\)[/tex]:
By taking the arcsine (inverse sine) of the value computed for [tex]\(\sin B\)[/tex], we get:
[tex]\[
B = \arcsin\left(\frac{18 \cdot \sin(109^\circ)}{28}\right)
\][/tex]
This calculation gives us [tex]\(\angle B_1\)[/tex]:
[tex]\[
B_1 \approx 37.43^\circ
\][/tex]
Since the sine function is positive in both the first and second quadrants, there is another possible angle [tex]\(\angle B_2\)[/tex], which is:
[tex]\[
B_2 = 180^\circ - B_1 \approx 180^\circ - 37.43^\circ = 142.57^\circ
\][/tex]
3. Step 3: Verify if [tex]\(\angle B_2\)[/tex] is possible:
We must ensure the sum of the angles in a triangle adds up to [tex]\(180^\circ\)[/tex].
Check the possibility of [tex]\(\angle B_1\)[/tex]:
[tex]\[
\angle A + \angle B_1 = 109^\circ + 37.43^\circ = 146.43^\circ < 180^\circ
\][/tex]
So, [tex]\(\angle B_1\)[/tex] is valid.
Check the possibility of [tex]\(\angle B_2\)[/tex]:
[tex]\[
\angle A + \angle B_2 = 109^\circ + 142.57^\circ = 251.57^\circ > 180^\circ
\][/tex]
Therefore, [tex]\(\angle B_2\)[/tex] is not valid.
4. Step 4: Calculate [tex]\(\angle C\)[/tex]:
If [tex]\(\angle B_1\)[/tex] is valid,
[tex]\[
\angle C_1 = 180^\circ - \angle A - \angle B_1 = 180^\circ - 109^\circ - 37.43^\circ = 33.57^\circ
\][/tex]
Since [tex]\(\angle B_2\)[/tex] is invalid, [tex]\(\angle C_2\)[/tex] does not exist:
[tex]\[
C_2 = \text{DNE}
\][/tex]
5. Step 5: Calculate [tex]\(c\)[/tex] using the Law of Sines:
For [tex]\(B_1\)[/tex] and [tex]\(C_1\)[/tex]:
[tex]\[
c_1 = \frac{a \cdot \sin(C_1)}{\sin(A)} = \frac{28 \cdot \sin(33.57^\circ)}{\sin(109^\circ)} \approx 16.37
\][/tex]
Since [tex]\(\angle B_2\)[/tex] does not exist, [tex]\(c_2\)[/tex] does not exist:
[tex]\[
c_2 = \text{DNE}
\][/tex]
So, the final answers are:
[tex]\[
\angle B_1 \approx 37.43^\circ, \quad \angle C_1 \approx 33.57^\circ, \quad c_1 \approx 16.37
\][/tex]
[tex]\[
\angle B_2 = \text{DNE}, \quad \angle C_2 = \text{DNE}, \quad c_2 = \text{DNE}
\][/tex]
