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Let's use the **Law of Sines** to solve the triangle. Here’s the given information:

- Angle \(A = 37^\circ\)  
- Side opposite \(A\) (i.e., \(a\)) = 9  
- Side opposite \(B\) (i.e., \(b\)) = 8  

### Step 1: Apply the Law of Sines  
The Law of Sines states:  
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]

We'll first find **angle \(B\)** using:
\[
\frac{\sin B}{b} = \frac{\sin A}{a}
\]

### Step 2: Solve for \(\sin B\)
\[
\sin B = \frac{b \cdot \sin A}{a} = \frac{8 \cdot \sin 37^\circ}{9}
\]

Using \(\sin 37^\circ \approx 0.6018\):
\[
\sin B = \frac{8 \cdot 0.6018}{9} \approx \frac{4.8144}{9} \approx 0.535
\]

Now, find \(B\):
\[
B = \sin^{-1}(0.535) \approx 32.41^\circ
\]

### Step 3: Find Angle \(C\)  
Since the sum of the angles in a triangle is \(180^\circ\):
\[
C = 180^\circ - A - B = 180^\circ - 37^\circ - 32.41^\circ \approx 110.59^\circ
\]

### Step 4: Find Side \(c\)  
Now, use the Law of Sines again to find side \(c\):
\[
\frac{c}{\sin C} = \frac{a}{\sin A}
\]

So:
\[
c = \frac{a \cdot \sin C}{\sin A} = \frac{9 \cdot \sin 110.59^\circ}{\sin 37^\circ}
\]

Using \(\sin 110.59^\circ \approx 0.9397\):
\[
c = \frac{9 \cdot 0.9397}{0.6018} \approx \frac{8.4573}{0.6018} \approx 14.05
\]

### Final Answers  
- \(B \approx 32.41^\circ\)  
- \(C \approx 110.59^\circ\)  
- \(c \approx 14.05\)

Would you like me to clarify or elaborate on any step?

---

Here are five follow-up questions:
1. How would the calculations change if angle \(A\) was obtuse?
2. How does the Law of Sines apply to non-right triangles?
3. Can you solve a triangle if only two sides and no angles are known?
4. What happens if \(\sin^{-1}(x)\) yields two possible angles?
5. What are some real-world applications of the Law of Sines?

**Tip:** Always check if the triangle is ambiguous (SSA case) when using the Law of Sines.

Use the Law of Sines to solve the triangle. Given: A = 37°, a = 9, b = 8. Find angles B, C, and side c. Round your answers to two decimal places.

Learn how to solve a triangle using the Law of Sines, given angle A = 37°, side a = 9, and side b = 8. Detailed step-by-step solution to find angles B, C, and side c. Suitable for students in Grades 10-12.

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