To solve the triangle in which angle $B$ is obtuse, we will use the Law of Sines and the information given for the triangle.

Given:

• $A = 44^\circ$
• $a = 9$
• $b = 11$

First, let's calculate angle $B$ when it is acute using the Law of Sines: $\frac{a}{\sin A} = \frac{b}{\sin B}$

$\frac{9}{\sin 44^\circ} = \frac{11}{\sin B}$

Solving for $\sin B$: $\sin B = \frac{11 \sin 44^\circ}{9}$

Calculating $\sin 44^\circ$ (using a calculator): $\sin 44^\circ \approx 0.694$

Now, substituting the value: $\sin B = \frac{11 \times 0.694}{9} \approx \frac{7.634}{9} \approx 0.848$

Therefore: $B \approx \sin^{-1}(0.848) \approx 58.1^\circ$

Given that there is a second triangle where $B$ is obtuse, we calculate the obtuse angle by: $B = 180^\circ - 58.1^\circ = 121.9^\circ$

Now, we find the remaining angle $C$: $C = 180^\circ - A - B$

For the obtuse triangle: $C = 180^\circ - 44^\circ - 121.9^\circ = 14.1^\circ$

Lastly, let's calculate the side $c$ using the Law of Sines: $\frac{c}{\sin C} = \frac{a}{\sin A}$

Rearranging to solve for $c$: $c = \frac{a \cdot \sin C}{\sin A}$

Substituting the known values: $\sin C = \sin 14.1^\circ \approx 0.244$ $c = \frac{9 \cdot 0.244}{0.694} \approx 3.16$

So, the measurements for the triangle in which $B$ is obtuse are:

• $A = 44^\circ$
• $B = 121.9^\circ$
• $C = 14.1^\circ$
• $a = 9$
• $b = 11$
• $c = 3.2$ (rounded to the nearest tenth)

Would you like more details or have any questions?

Here are some related questions to expand your understanding:

1. What is the Law of Sines and how is it used in solving triangles?
2. How do you determine if an angle in a triangle is obtuse or acute?
3. Can the Law of Cosines be used to solve this triangle? If so, how?
4. How would the solution differ if side $a$ was longer than side $b$?
5. What are the properties of an obtuse triangle?
6. How do you calculate the area of a triangle given two sides and the included angle?
7. How can you verify the accuracy of your calculations in trigonometric problems?
8. What are some real-life applications of solving triangles?

Tip: When solving triangles using trigonometric functions, ensure your calculator is set to the correct mode (degrees or radians) based on the given angles.