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To solve for \( b \) using the Law of Sines, we first need to set up the proportion. The Law of Sines states:

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

Given:
- \( a = 18 \)
- \( A = 28^\circ \)
- \( B = 83^\circ \)

We need to find \( b \).

First, we need to find the third angle \( C \) of the triangle:
\[
C = 180^\circ - A - B = 180^\circ - 28^\circ - 83^\circ = 69^\circ
\]

Now, using the Law of Sines:
\[
\frac{a}{\sin A} = \frac{b}{\sin B}
\]

Substituting the given values:
\[
\frac{18}{\sin 28^\circ} = \frac{b}{\sin 83^\circ}
\]

Solving for \( b \):
\[
b = \frac{18 \sin 83^\circ}{\sin 28^\circ}
\]

Now, we need to calculate the values of the sine functions:
\[
\sin 83^\circ \approx 0.9925
\]
\[
\sin 28^\circ \approx 0.4695
\]

So:
\[
b = \frac{18 \times 0.9925}{0.4695} \approx \frac{17.865}{0.4695} \approx 38.1
\]

Therefore, the value of \( b \) rounded to the nearest tenth is:
\[
b \approx 38.1
\]

Would you like more details or have any questions?

Here are 8 related questions for further exploration:
1. How does the Law of Cosines apply to this triangle?
2. What are the steps to solve for the third side using the Law of Cosines?
3. Can we solve for the third side using a different method?
4. What are the general properties of a triangle in terms of angle sum?
5. How do you find the area of this triangle?
6. What is the importance of the Law of Sines in solving real-world problems?
7. How do you verify the solution obtained using the Law of Sines?
8. What are the limitations of the Law of Sines?

**Tip:** Always double-check your angle measures and sine values for accuracy when using trigonometric laws to solve for unknown sides or angles in a triangle.

Learn how to solve for side b in a triangle using the Law of Sines with detailed step-by-step calculations. This problem involves trigonometric laws, angle calculations, and real-world application. Suitable for high school students studying trigonometry.

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