Is the function `f(x, y) = sin x + cos y` homogeneous ?

Answer

To determine if the function \( f(x, y) = \sin x + \cos y \) is homogeneous, we need to check if it satisfies the condition for homogeneity.

### Step-by-Step Solution:

1. **Definition of Homogeneous Function**: A function \( f(x, y) \) is said to be homogeneous of degree \( n \) if, for any scalar \( \lambda \), the following holds:
\[ f(\lambda x, \lambda y) = \lambda^n f(x, y) \]
This means that when we scale both \( x \) and \( y \) by \( \lambda \), the function scales by \( \lambda^n \).

2. **Substituting \( \lambda x \) and \( \lambda y \)**: We need to evaluate \( f(\lambda x, \lambda y) \):
\[ f(\lambda x, \lambda y) = \sin(\lambda x) + \cos(\lambda y) \]

3. **Comparison with the Original Function**: Now we need to compare \( f(\lambda x, \lambda y) \) with \( \lambda^n f(x, y) \):
\[ f(x, y) = \sin x + \cos y \]
Therefore, \( \lambda^n f(x, y) = \lambda^n (\sin x + \cos y) \).

4. **Analyzing the Result**: The expression \( \sin(\lambda x) + \cos(\lambda y) \) does not simplify to \( \lambda^n (\sin x + \cos y) \) for any integer \( n \). This is because sine and cosine functions do not exhibit a simple scaling property when their arguments are scaled.

5. **Conclusion**: Since we cannot express \( f(\lambda x, \lambda y) \) in the form of \( \lambda^n f(x, y) \), the function \( f(x, y) = \sin x + \cos y \) is not homogeneous.

### Final Answer:
The function \( f(x, y) = \sin x + \cos y \) is **not homogeneous**.