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The problem you provided involves trigonometry and asks to solve for \( \sin(\alpha) \), given certain conditions. Here's a breakdown of what I see in the image:

**Problem (BT1)**:
- Given \( \tan(\alpha) = 3 \) 
- The range for \( \alpha \) is \( \pi < \alpha < \frac{3\pi}{2} \) 
- The task is to find \( \sin(\alpha) \).

### Solution:
1. **Understanding the range**: 
   The range \( \pi < \alpha < \frac{3\pi}{2} \) means that \( \alpha \) is in the third quadrant, where:
   - \( \sin(\alpha) \) is negative
   - \( \cos(\alpha) \) is negative
   - \( \tan(\alpha) = \frac{\sin(\alpha)}{\cos(\alpha)} \) is positive

2. **Using the given \( \tan(\alpha) = 3 \)**:
   We know that:
   \[
   \tan(\alpha) = \frac{\sin(\alpha)}{\cos(\alpha)} = 3
   \]
   This implies:
   \[
   \frac{\sin(\alpha)}{\cos(\alpha)} = 3 \quad \Rightarrow \quad \sin(\alpha) = 3\cos(\alpha)
   \]

3. **Pythagorean identity**:
   From the identity \( \sin^2(\alpha) + \cos^2(\alpha) = 1 \), substitute \( \sin(\alpha) = 3\cos(\alpha) \):
   \[
   (3\cos(\alpha))^2 + \cos^2(\alpha) = 1
   \]
   Simplifying:
   \[
   9\cos^2(\alpha) + \cos^2(\alpha) = 1
   \]
   \[
   10\cos^2(\alpha) = 1 \quad \Rightarrow \quad \cos^2(\alpha) = \frac{1}{10}
   \]
   \[
   \cos(\alpha) = -\frac{1}{\sqrt{10}} \quad (\text{since } \alpha \text{ is in the third quadrant, } \cos(\alpha) \text{ is negative})
   \]

4. **Find \( \sin(\alpha) \)**:
   Now, using \( \sin(\alpha) = 3\cos(\alpha) \):
   \[
   \sin(\alpha) = 3 \times -\frac{1}{\sqrt{10}} = -\frac{3}{\sqrt{10}}
   \]
   Therefore, \( \sin(\alpha) = -\frac{3}{\sqrt{10}} \).

### Final Answer:
\[
\sin(\alpha) = -\frac{3}{\sqrt{10}}
\]

Would you like further details on any part of this solution?

### Relative questions:
1. How do you derive the value of \( \cos(\alpha) \) given \( \tan(\alpha) \)?
2. What are the properties of trigonometric functions in the third quadrant?
3. How does the Pythagorean identity help in solving for \( \cos(\alpha) \) or \( \sin(\alpha) \)?
4. Can you find the exact value of \( \cos(\alpha) \) if \( \tan(\alpha) \) is given and \( \alpha \) is in the first quadrant?
5. What happens if \( \alpha \) were in the fourth quadrant? Would \( \sin(\alpha) \) be positive or negative?

### Tip:
Always consider the sign of trigonometric functions based on the quadrant when solving such problems. This ensures that your final values are accurate according to the angle's position.

Cho tan(α) = 3, π < α < (3π)/2. Tính sin(α)

Learn how to solve for sin(α) when tan(α) = 3 and α is between π and (3π)/2. This solution uses trigonometric identities like the Pythagorean identity and considers the properties of angles in the third quadrant. Suitable for students in Grades 10-12.

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