To determine how many distinct triangles can be formed with the given information [tex]\( m \angle A = 75^{\circ} \)[/tex], [tex]\( a = 2 \)[/tex], and [tex]\( b = 3 \)[/tex], we will use the Law of Sines. The Law of Sines states that:
[tex]\[
\frac{\sin A}{a} = \frac{\sin B}{b}
\][/tex]
Step-by-Step Solution:
1. We are given:
- [tex]\( \angle A = 75^\circ \)[/tex]
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = 3 \)[/tex]
2. Substitute these values into the Law of Sines:
[tex]\[
\frac{\sin(75^\circ)}{2} = \frac{\sin(B)}{3}
\][/tex]
3. Solving for [tex]\( \sin(B) \)[/tex]:
[tex]\[
\sin(B) = 3 \cdot \frac{\sin(75^\circ)}{2}
\][/tex]
4. We know that [tex]\( \sin(75^\circ) \approx 0.9659 \)[/tex]:
[tex]\[
\sin B = 3 \cdot \frac{0.9659}{2} = 3 \cdot 0.48295 = 1.44885
\][/tex]
5. Given the range of the sine function is [tex]\(-1 \leq \sin x \leq 1\)[/tex], we see that [tex]\( \sin B = 1.44885 \)[/tex] is not a valid value. Therefore, there is no angle [tex]\( B \)[/tex] that satisfies this equation.
Conclusion:
Since [tex]\( \sin(B) \)[/tex] exceeds the maximum possible value of 1, it indicates that no valid angle [tex]\( B \)[/tex] can be formed and thus no triangle can be constructed with the given parameters.
Hence, no triangles can be formed.
