QuestionDetermine the number of triangles $A B C$ possible with the given parts. $$A=42.8^{\circ} \quad a=8.9 \quad b=10.1$$
How many possible solutions does this triangle have? $\square$

Studdy Solution
Determine the possible values for angle $B$ using $\sin B$:
Since $\sin B \approx 0.774$, angle $B$ could be:
$$B = \sin^{-1}(0.774) \approx 50.8^\circ$$
Check for the possibility of a second triangle using the supplementary angle:
$$B' = 180^\circ - 50.8^\circ = 129.2^\circ$$
Verify if both angles $B$ and $B'$ lead to valid triangles:
1. For $B = 50.8^\circ$, the sum of angles $A + B = 42.8^\circ + 50.8^\circ = 93.6^\circ$, which is less than $180^\circ$. This is a valid triangle.
2. For $B' = 129.2^\circ$, the sum of angles $A + B' = 42.8^\circ + 129.2^\circ = 172^\circ$, which is also less than $180^\circ$. This is another valid triangle.
There are two possible solutions for the triangle $\triangle ABC$.
$$\boxed{2}$$