To determine how many possible triangles can be formed given that [tex]\( m \angle A = \frac{\pi}{6} \)[/tex], [tex]\( c = 18 \)[/tex], and [tex]\( a = 9 \)[/tex], we can follow these steps:
1. Convert the given angle into radians:
Given [tex]\( m \angle A = \frac{\pi}{6} \)[/tex].
2. Determine [tex]\( \sin \angle A \)[/tex]:
Since [tex]\( \angle A = \frac{\pi}{6} \)[/tex], we have:
[tex]\[
\sin \angle A = \sin \left( \frac{\pi}{6} \right) = \frac{1}{2}.
\][/tex]
3. Apply the Law of Sines:
Using the Law of Sines, we have:
[tex]\[
\frac{a}{\sin(\angle A)} = \frac{c}{\sin(\angle B)}.
\][/tex]
Plugging in the values [tex]\( a = 9 \)[/tex] and [tex]\( \sin(\angle A) = \frac{1}{2} \)[/tex], we get:
[tex]\[
\frac{9}{\frac{1}{2}} = \frac{18}{\sin(\angle B)}.
\][/tex]
Simplifying this, we obtain:
[tex]\[
18 = \frac{18}{\sin(\angle B)}.
\][/tex]
4. Solve for [tex]\( \sin(\angle B) \)[/tex]:
[tex]\[
\sin(\angle B) = \frac{18}{18} = 1.
\][/tex]
5. Check the value of [tex]\( \sin(\angle B) \)[/tex]:
The value [tex]\( \sin(\angle B) = 1 \)[/tex] means that:
[tex]\[
\angle B = \frac{\pi}{2}.
\][/tex]
6. Determine the possible values for [tex]\( \angle B \)[/tex]:
Since [tex]\(\sin(\angle B) = \frac{1}{4}\)[/tex], we need to find the possible angles [tex]\( \angle B \)[/tex] that satisfy this equation. [tex]\(\sin(\angle B) = 0.25\)[/tex] can result in two possible angles within the valid range:
- [tex]\( \angle B \approx 0.2527 \text{ radians} \)[/tex],
- [tex]\( \angle B \approx \pi - 0.2527 \text{ radians} \)[/tex].
7. Calculate the number of possible triangles:
Both angles [tex]\( \angle B \approx 0.2527 \text{ radians} \)[/tex] and [tex]\( \pi - 0.2527 \text{ radians} \)[/tex] will, along with [tex]\( \angle A = \frac{\pi}{6} \)[/tex], form valid triangles because the third angle will also satisfy the triangle angle sum property (sum up to [tex]\(\pi\)[/tex]).
Therefore, there are [tex]\(2\)[/tex] triangles that can be formed with the given conditions.
Answer: 2 triangles.
