First, we need to determine if the given triangle with side lengths [tex]\(a = 22\)[/tex], [tex]\(A = 117^\circ\)[/tex], and [tex]\(b = 25\)[/tex] can form a valid triangle, and if so, how many possible solutions there are.
We can use the Law of Sines to find the possible values for angle [tex]\(B\)[/tex]:
[tex]\[
\sin(A) = \sin(117^\circ)
\][/tex]
Next, according to Law of Sines:
[tex]\[
\frac{\sin(A)}{a} = \frac{\sin(B)}{b}
\][/tex]
Rewriting this equation, we have:
[tex]\[
\sin(B) = \frac{b \cdot \sin(A)}{a}
\][/tex]
We must ensure that the value calculated for [tex]\(\sin(B)\)[/tex] falls within the range of possible sine values, which is [tex]\(-1\)[/tex] to [tex]\(1\)[/tex]. If it falls outside this range, it means no solution is possible.
Assuming [tex]\(\sin(B)\)[/tex] is valid, we solve for [tex]\(B\)[/tex]:
1. First, compute the principal value [tex]\(B_1\)[/tex], which is given by the arcsine function.
2. Additionally, consider the supplementary angle [tex]\(B_2\)[/tex], because the sine function is positive in both the first and second quadrants for one complete cycle [tex]\(0^\circ\)[/tex] to [tex]\(180^\circ\)[/tex]. Thus:
[tex]\[
B_2 = 180^\circ - B_1
\][/tex]
Once we have the two possible values for [tex]\(B\)[/tex], we need to compute the corresponding values for angle [tex]\(C\)[/tex]:
[tex]\[
C_1 = 180^\circ - A - B_1 \\
C_2 = 180^\circ - A - B_2
\][/tex]
For the triangle to be a valid one, [tex]\(C\)[/tex] must be positive. We then check the validity of [tex]\(C_1\)[/tex] and [tex]\(C_2\)[/tex]:
- If [tex]\(C_1 > 0\)[/tex], then it's a valid angle combination.
- If [tex]\(C_2 > 0\)[/tex], then it's another valid angle combination.
Counting the valid combinations will give us the number of possible solutions.
The detailed checking process yields:
[tex]\[
\text{Number of solutions} = 1
\][/tex]
