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The problem asks to solve triangle \( ABC \) where:

- \( a = 15 \)
- \( c = 20 \)
- \( \beta = 60^\circ \)

We are required to find the unknowns: \( b \), angle \( \alpha \), and angle \( \gamma \).

### Step-by-step solution:

1. **Find angle \( \gamma \)**:
   Since the sum of the angles in a triangle is always \( 180^\circ \), we can calculate \( \gamma \) as:
   \[
   \gamma = 180^\circ - \beta - \alpha
   \]
   However, we first need to find \( \alpha \).

2. **Use the Law of Cosines to find \( b \)**:
   The Law of Cosines states:
   \[
   b^2 = a^2 + c^2 - 2ac \cdot \cos(\beta)
   \]
   Substituting the known values:
   \[
   b^2 = 15^2 + 20^2 - 2 \cdot 15 \cdot 20 \cdot \cos(60^\circ)
   \]
   Since \( \cos(60^\circ) = 0.5 \), we get:
   \[
   b^2 = 225 + 400 - 2 \cdot 15 \cdot 20 \cdot 0.5
   \]
   \[
   b^2 = 625 - 300 = 325
   \]
   Taking the square root:
   \[
   b = \sqrt{325} \approx 18.03
   \]

3. **Use the Law of Sines to find angle \( \alpha \)**:
   The Law of Sines states:
   \[
   \frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} = \frac{c}{\sin(\gamma)}
   \]
   Using \( \frac{a}{\sin(\alpha)} = \frac{b}{\sin(\beta)} \):
   \[
   \frac{15}{\sin(\alpha)} = \frac{18.03}{\sin(60^\circ)}
   \]
   Since \( \sin(60^\circ) = \frac{\sqrt{3}}{2} \approx 0.866 \), we get:
   \[
   \frac{15}{\sin(\alpha)} = \frac{18.03}{0.866}
   \]
   Solving for \( \sin(\alpha) \):
   \[
   \sin(\alpha) = \frac{15 \cdot 0.866}{18.03} \approx 0.721
   \]
   Thus:
   \[
   \alpha = \arcsin(0.721) \approx 46.07^\circ
   \]

4. **Find angle \( \gamma \)**:
   Now, using the fact that the angles in a triangle sum to \( 180^\circ \):
   \[
   \gamma = 180^\circ - 60^\circ - 46.07^\circ \approx 73.93^\circ
   \]

### Final solution:
- \( b \approx 18.03 \)
- \( \alpha \approx 46.07^\circ \)
- \( \gamma \approx 73.93^\circ \)

Would you like further clarifications or details on any step?

Here are 5 questions related to this problem:
1. What are the differences between the Law of Sines and the Law of Cosines?
2. How can the Law of Cosines help when all three sides are known?
3. What happens when one of the angles in a triangle is obtuse?
4. Can you solve a triangle if only two angles are given?
5. What is the relation between angles and sides in a triangle?

**Tip**: When solving triangles, always check if it's appropriate to use the Law of Sines or Cosines based on the given information.

Solve triangle ABC with a = 15, c = 20 and β = 60° using usual notations.

Learn how to solve triangle ABC with sides a = 15, c = 20, and angle β = 60°. This detailed solution covers the Law of Cosines and Law of Sines to find the unknown side and angles, suitable for Grade 11-12 students.

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