Given:
B = 30°, a = 4, b = 3
We will solve the triangle to find how many triangles exist that fit the given data
We will use the sine rule to find the angle A as follows:
[tex]\begin{gathered} \frac{a}{sin(A)}=\frac{b}{sin(B)} \\ \\ \frac{4}{sin(A)}=\frac{3}{sin(30)} \\ \\ sin(A)=\frac{4}{3}sin(30)=\frac{2}{3} \\ \end{gathered}[/tex]
So, the measure of angle A will be as follows:
[tex]A=sin^{-1}(\frac{2}{3})=41.81\degree,or,138.19\degree[/tex]
Now, we will find the measure of angle C using the fact that the sum of the angles = 180
[tex]\begin{gathered} C=180-(A+B) \\ A=41.81\degree\rightarrow C=180-(30+41.81)=108.19\operatorname{\degree} \\ A=138.19\operatorname{\degree}\rightarrow C=180-(30+138.19)=11.81\operatorname{\degree} \end{gathered}[/tex]
So, there are two triangles that can fit the given data
So, the answer will be Two
