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The problem is asking to solve a triangle \( \triangle ABC \) where \( c = 28 \), \( b = 34 \), and \( \angle C = 36^\circ \). We are tasked with finding the other two angles, \( A \) and \( B \), and the remaining side \( a \).

### Step-by-step solution:

1. **Using the Law of Sines**:
   \[
   \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
   \]
   We know \( c = 28 \), \( b = 34 \), and \( \angle C = 36^\circ \), so we can first find angle \( B \).

2. **Find angle \( B \) using the Law of Sines**:
   \[
   \frac{b}{\sin B} = \frac{c}{\sin C}
   \]
   \[
   \frac{34}{\sin B} = \frac{28}{\sin 36^\circ}
   \]
   Solve for \( \sin B \):
   \[
   \sin B = \frac{34 \cdot \sin 36^\circ}{28}
   \]

   Let's compute \( \sin B \) and find \( B \).

I'll calculate this.We found that angle \( B \approx 45.5^\circ \).

3. **Find angle \( A \) using the angle sum property of triangles**:
   Since the sum of angles in any triangle is \( 180^\circ \), we can find \( A \):
   \[
   A = 180^\circ - C - B
   \]
   Substituting \( C = 36^\circ \) and \( B = 45.5^\circ \):
   \[
   A = 180^\circ - 36^\circ - 45.5^\circ
   \]

Let me calculate \( A \).We found that angle \( A \approx 98.5^\circ \).

4. **Find side \( a \) using the Law of Sines**:
   Now, we use the Law of Sines again to find side \( a \):
   \[
   \frac{a}{\sin A} = \frac{c}{\sin C}
   \]
   Solving for \( a \):
   \[
   a = \frac{c \cdot \sin A}{\sin C}
   \]

Let me compute side \( a \).The length of side \( a \approx 47.1 \).

### Final Solution:
- Angle \( A \approx 98.5^\circ \)
- Angle \( B \approx 45.5^\circ \)
- Side \( a \approx 47.1 \)

Would you like further details or explanations on any part of the process?

Here are 5 related questions to expand on this topic:
1. How would the solution change if \( \angle C \) was an obtuse angle?
2. Can we use the Law of Cosines in this problem? Why or why not?
3. How can the ambiguous case of the Law of Sines affect the solution?
4. What happens if side \( b \) were smaller than \( c \) in this scenario?
5. How would you approach solving this if all sides but no angles were given?

**Tip**: When solving triangles, always check whether there might be multiple solutions, especially when using the Law of Sines.

Consider a triangle ABC like the one below. Suppose that c = 28, b = 34, and C = 36°. Solve the triangle.

Solve triangle ABC using the Law of Sines, given c = 28, b = 34, and C = 36°. Detailed solution includes angle B, angle A, and side a, with key steps and explanations.

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