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The image shows a mathematical problem involving trigonometric functions. The question asks the following:

**Problem:**
- Given the inequality \( \frac{3\pi}{2} < \theta < 2\pi \) and \( \tan(\theta) = -3 \), determine the values of \( \sin(\theta) \) and \( \cos(\theta) \).

### Solution Outline:

1. **Understanding the interval**: 
   \( \frac{3\pi}{2} < \theta < 2\pi \) places the angle \( \theta \) in the fourth quadrant, where:
   - \( \sin(\theta) \) is negative
   - \( \cos(\theta) \) is positive
   - \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \) is negative (as expected from the given \( \tan(\theta) = -3 \)).

2. **Using the identity for tangent**:
   \[
   \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = -3
   \]
   This implies that \( \sin(\theta) = -3\cos(\theta) \).

3. **Using the Pythagorean identity**:
   \[
   \sin^2(\theta) + \cos^2(\theta) = 1
   \]
   Substitute \( \sin(\theta) = -3\cos(\theta) \) into this identity:
   \[
   (-3\cos(\theta))^2 + \cos^2(\theta) = 1
   \]
   Simplifying:
   \[
   9\cos^2(\theta) + \cos^2(\theta) = 1
   \]
   \[
   10\cos^2(\theta) = 1
   \]
   \[
   \cos^2(\theta) = \frac{1}{10}
   \]
   \[
   \cos(\theta) = \frac{1}{\sqrt{10}}
   \]
   Since \( \theta \) is in the fourth quadrant, \( \cos(\theta) \) is positive:
   \[
   \cos(\theta) = \frac{1}{\sqrt{10}}
   \]

4. **Finding \( \sin(\theta) \)**:
   From \( \sin(\theta) = -3\cos(\theta) \), we substitute the value of \( \cos(\theta) \):
   \[
   \sin(\theta) = -3 \times \frac{1}{\sqrt{10}} = -\frac{3}{\sqrt{10}}
   \]

Thus, the solutions are:
\[
\sin(\theta) = -\frac{3}{\sqrt{10}}, \quad \cos(\theta) = \frac{1}{\sqrt{10}}
\]

Would you like more details, or do you have any questions?

### Related Questions:
1. How do you determine the quadrant based on angle inequalities?
2. What is the significance of the signs of sine and cosine in different quadrants?
3. Can you explain the tangent function's relationship to sine and cosine?
4. How can the Pythagorean identity help solve for sine and cosine in general?
5. What other methods can be used to find sine and cosine given a trigonometric ratio?

**Tip**: Always remember to check the quadrant to assign the correct signs to trigonometric functions.

Jika \( \frac{3\pi}{2} < 0 < 2\pi \) dan \( \tan(\theta) = -3 \), tentukan nilai \( \sin(\theta) \) dan \( \cos(\theta) \).

Solve for \( \sin(\theta) \) and \( \cos(\theta) \) given \( \tan(\theta) = -3 \) and \( \theta \) is in the fourth quadrant (\( \frac{3\pi}{2} < \theta < 2\pi \)). Learn how to use the tangent function and the Pythagorean identity to determine the values of sine and cosine.

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