To solve for all possible triangles that satisfy the given conditions using the Law of Sines, we are given the following information:

• $b = 45$ (side opposite ∠B)
• $c = 43$ (side opposite ∠C)
• $\angle C = 36^\circ$

We can apply the Law of Sines, which states:

$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$

1. First, we use the known values for $b$, $c$, and $\angle C$ to solve for $\sin B$:

$\frac{\sin B}{45} = \frac{\sin 36^\circ}{43}$

$\sin B = 45 \cdot \frac{\sin 36^\circ}{43}$

Let's calculate $\sin B$.We have:

$\sin B \approx 0.6151$

Now, we can find $\angle B$ by taking the inverse sine of this value:

$\angle B = \arcsin(0.6151)$

Let’s calculate $\angle B$.We have:

$\angle B \approx 38.0^\circ$

Next, we can calculate $\angle A$ using the fact that the sum of the angles in a triangle is $180^\circ$:

$\angle A = 180^\circ - \angle B - \angle C$

Let’s calculate $\angle A$.We have:

$\angle A \approx 106.0^\circ$

This gives us one possible triangle. Now, we check if there is a second possible triangle by considering the fact that $\sin B$ can also correspond to $180^\circ - \angle B$. This is because sine has the same value for an angle and its supplement.

So, we calculate the second possible value for $\angle B$:

$\angle B_2 = 180^\circ - \angle B$

Let’s calculate $\angle B_2$.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, here's the process you can follow to find the second possible triangle manually:

1. Calculate the second possible angle $B_2$: $\angle B_2 = 180^\circ - \angle B$ Substituting $\angle B \approx 38^\circ$, we get: $\angle B_2 = 180^\circ - 38^\circ = 142^\circ$

2. Check if $\angle B_2 + \angle C$ is less than $180^\circ$: $\angle B_2 + \angle C = 142^\circ + 36^\circ = 178^\circ$ Since the sum is less than $180^\circ$, a second triangle is possible.

3. Calculate $\angle A_2$: $\angle A_2 = 180^\circ - \angle B_2 - \angle C$ $\angle A_2 = 180^\circ - 142^\circ - 36^\circ = 2^\circ$

Thus, there are two possible triangles:

• First triangle: $\angle A_1 \approx 106.0^\circ$, $\angle B_1 \approx 38.0^\circ$, $\angle C = 36^\circ$
• Second triangle: $\angle A_2 \approx 2.0^\circ$, $\angle B_2 \approx 142^\circ$, $\angle C = 36^\circ$

Let me know if you'd like further details!

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Follow-up questions:

1. How does the Law of Sines help determine ambiguous triangles?
2. What is the general form of the Law of Sines?
3. Why do we check for a second solution using $180^\circ - \angle B$?
4. How can you determine if a second triangle is possible?
5. What happens if $\sin B$ is greater than 1?

Tip: Always check for a second triangle when solving with the Law of Sines, as it often leads to ambiguous cases.