To determine how many solutions a triangle with sides [tex]\(a = 42\)[/tex], [tex]\(b = 34\)[/tex], and angle [tex]\(A = 117^\circ\)[/tex] has, we utilize the Law of Sines and triangle properties. Here's a step-by-step solution:
1. Understanding the Law of Sines:
[tex]\[
\frac{a}{\sin(A)} = \frac{b}{\sin(B)}
\][/tex]
We need to find angle [tex]\(B\)[/tex] and analyze the number of valid triangles.
2. Calculate [tex]\(\sin(A)\)[/tex]:
[tex]\[
A = 117^\circ
\][/tex]
Convert this angle to radians for calculation:
[tex]\[
A_{\text{radians}} = \frac{117 \pi}{180}
\][/tex]
Calculate [tex]\(\sin(A)\)[/tex]:
[tex]\[
\sin(A) = \sin(117^\circ)
\][/tex]
3. Using the Law of Sines, solve for [tex]\(\sin(B)\)[/tex]:
[tex]\[
\sin(B) = \frac{b \cdot \sin(A)}{a}
\][/tex]
4. Given values:
[tex]\[
a = 42
\][/tex]
[tex]\[
b = 34
\][/tex]
5. Evaluate [tex]\(\sin(B)\)[/tex]:
[tex]\[
\sin(B) = \frac{34 \cdot \sin(117^\circ)}{42}
\][/tex]
6. Check the value of [tex]\(\sin(B)\)[/tex]:
- If [tex]\(\sin(B) > 1\)[/tex] or [tex]\(\sin(B) < -1\)[/tex], there is no valid triangle since [tex]\(\sin\)[/tex] of an angle cannot be outside the range [tex]\([-1, 1]\)[/tex].
- If [tex]\(\sin(B)\)[/tex] is within the range [tex]\([-1, 1]\)[/tex], proceed to calculate [tex]\(B\)[/tex].
7. Two possibilities for angle [tex]\(B\)[/tex] when [tex]\(\sin(B)\)[/tex] is valid:
[tex]\[
B = \arcsin(\sin(B))
\][/tex]
[tex]\[
B' = 180^\circ - B
\][/tex]
Since the sum of the angles in a triangle must be [tex]\(180^\circ\)[/tex]:
- Check if [tex]\(B + 117^\circ < 180^\circ\)[/tex]: This validates [tex]\(B\)[/tex] as a solution.
- Check if [tex]\(B' + 117^\circ < 180^\circ\)[/tex]: This validates [tex]\(B'\)[/tex] as a second possible solution.
Conclusion:
After evaluating these conditions, we find that there are indeed 2 solutions. Thus, the number of solutions a triangle with the given dimensions has is:
[tex]\[
2
\][/tex]
