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The problem requires matching functions \( f(x) = \cos(kx) \) and \( g(x) = \sin(kx) \) to certain transformation descriptions.

Given:
- \( f(x) = \cos(kx) \)
- \( g(x) = \sin(kx) \)

The task is to match each description in List I to the correct entries in List II.

**List I:**
1. \( k \) is an integer
2. Transpose of \( (x, y) \)
3. Transpose of \( (-x, y) \)
4. Transpose of \( \left( \frac{x}{k}, y \right) \)

**List II:**
- \( k = m \pi \) 
- \( (k, n) \)
- \( \left( \frac{1}{x}, \cos(kx) \right) \)
- \( k = 2m \pi \)

Let's interpret the information:

### Analysis:

1. **\( k \) is an integer:**
   - This implies that \( k = m \pi \), where \( m \) is an integer, is a possible match.
  
2. **Transpose of \( (x, y) \):**
   - This means switching the roles of \( x \) and \( y \). The likely match for this transformation could be \( (k, n) \) from List II, indicating that \( x \) could represent \( k \) and \( y \) could represent \( n \).

3. **Transpose of \( (-x, y) \):**
   - This implies a transformation involving \( x \to -x \). No clear match can be identified for this from the given pairs.

4. **Transpose of \( \left( \frac{x}{k}, y \right) \):**
   - This transformation suggests scaling \( x \) by \( \frac{1}{k} \). The corresponding match is \( \left( \frac{1}{x}, \cos(kx) \right) \).

### Matching:
From the analysis above, we can match the items as follows:

1. \( k \) is an integer \(\rightarrow\) \( k = m\pi \) (Option II)
2. Transpose of \( (x, y) \) \(\rightarrow\) \( (k, n) \) (Option I)
3. Transpose of \( (-x, y) \) \(\rightarrow\) \( k = 2m\pi \) (Option IV)
4. Transpose of \( \left( \frac{x}{k}, y \right) \) \(\rightarrow\) \( \left( \frac{1}{x}, \cos(kx) \right) \) (Option III)

### Answer:
1. \( k \) is an integer \(\rightarrow\) \( k = m\pi \)
2. Transpose of \( (x, y) \) \(\rightarrow\) \( (k, n) \)
3. Transpose of \( (-x, y) \) \(\rightarrow\) \( k = 2m\pi \)
4. Transpose of \( \left( \frac{x}{k}, y \right) \) \(\rightarrow\) \( \left( \frac{1}{x}, \cos(kx) \right) \)

Would you like further details or have any questions about the solution?

### Follow-up Questions:

1. What does it mean for \( k \) to be an integer in trigonometric functions?
2. How does the transformation \( (x, y) \rightarrow (-x, y) \) affect the function graphically?
3. Can you explain why the transformation \( \left( \frac{x}{k}, y \right) \) matches \( \left( \frac{1}{x}, \cos(kx) \right) \)?
4. How do transformations impact the period and amplitude of trigonometric functions?
5. How would the function \( g(x) = \sin(kx) \) change if \( k = 2m \pi \)?

### Tip:
To better understand function transformations, try plotting the original and transformed functions on a graph to see the changes visually.

Explore how to match trigonometric functions \( f(x) = \cos(kx) \) and \( g(x) = \sin(kx) \) to given transformation descriptions. This problem involves understanding function transformations and their effects on trigonometric graphs. Suitable for advanced high school students and learners interested in trigonometry.

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