This problem involves solving a triangle given two sides ($a = 8.2 \, \text{m}$, $b = 10.7 \, \text{m}$) and one angle ($A = 44.5^\circ$). We can solve this triangle using the Law of Sines, as it relates angles to their opposite sides. However, we must first check whether there is one solution, two solutions, or no solution. This is known as the ambiguous case (SSA case) in trigonometry.

Step 1: Use the Law of Sines

The Law of Sines is expressed as: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ We can solve for angle $B$ first. Rewriting the Law of Sines for $B$: $\sin B = \frac{b \sin A}{a}$ Substitute the known values: $\sin B = \frac{10.7 \cdot \sin 44.5^\circ}{8.2}$

Step 2: Calculate $\sin B$

First, calculate $\sin 44.5^\circ$, and then solve for $\sin B$: $\sin 44.5^\circ \approx 0.6986$ $\sin B = \frac{10.7 \cdot 0.6986}{8.2} \approx 0.9114$

Step 3: Check for possible solutions

Since $\sin B \approx 0.9114$, and the sine of an angle can only be between -1 and 1, this indicates that angle $B$ exists.

Now, we find $B$ using the inverse sine function: $B = \sin^{-1}(0.9114) \approx 65.6^\circ$

Since $\sin B$ is positive, there are two possible solutions for $B$ because $\sin(\theta) = \sin(180^\circ - \theta)$. Thus:

• The first possible value of $B$ is $65.6^\circ$.
• The second possible value of $B$ is $180^\circ - 65.6^\circ = 114.4^\circ$.

Step 4: Check the sum of angles

If $B = 65.6^\circ$, then angle $C$ can be found by using the fact that the sum of angles in a triangle is $180^\circ$: $C = 180^\circ - A - B = 180^\circ - 44.5^\circ - 65.6^\circ = 69.9^\circ$

If $B = 114.4^\circ$, then: $C = 180^\circ - A - B = 180^\circ - 44.5^\circ - 114.4^\circ = 21.1^\circ$

Step 5: Solve for side $c$

Using the Law of Sines again: $\frac{c}{\sin C} = \frac{a}{\sin A}$ Thus: $c = \frac{a \cdot \sin C}{\sin A}$

For the first case, where $C = 69.9^\circ$: $c = \frac{8.2 \cdot \sin 69.9^\circ}{\sin 44.5^\circ} \approx \frac{8.2 \cdot 0.9362}{0.6986} \approx 10.98 \, \text{m}$

For the second case, where $C = 21.1^\circ$: $c = \frac{8.2 \cdot \sin 21.1^\circ}{\sin 44.5^\circ} \approx \frac{8.2 \cdot 0.3595}{0.6986} \approx 4.22 \, \text{m}$

Final Answer:

There are two possible solutions for the triangle:

1. First solution:

   • $B = 65.6^\circ$
   • $C = 69.9^\circ$
   • $c = 11.0 \, \text{m}$

2. Second solution:

   • $B = 114.4^\circ$
   • $C = 21.1^\circ$
   • $c = 4.2 \, \text{m}$

Would you like further details or clarifications?

Related Questions:

1. What are the conditions for having two solutions in a triangle with SSA?
2. How does the Law of Sines apply to ambiguous cases in trigonometry?
3. Can there ever be no solution for a triangle with SSA? Under what conditions?
4. How do you use the Law of Cosines to solve triangles?
5. What are the possible ranges for the sine of an angle in a triangle?

Tip:

When solving SSA triangles, always check the possibility of two solutions by considering both $\theta$ and $180^\circ - \theta$.