Evaluate `sin{ n pi+(-1)^(n) (pi)/(4)`, where n is an integer.

Answer

To evaluate the expression \( \sin\left(n \pi + (-1)^{n} \frac{\pi}{4}\right) \), where \( n \) is an integer, we can break it down step by step.

### Step 1: Understand the expression
The expression consists of two parts: \( n \pi \) and \( (-1)^{n} \frac{\pi}{4} \). The term \( n \pi \) represents integer multiples of \( \pi \), and the term \( (-1)^{n} \frac{\pi}{4} \) will alternate between \( \frac{\pi}{4} \) and \( -\frac{\pi}{4} \) depending on whether \( n \) is even or odd.

### Step 2: Evaluate the sine function
Using the sine angle addition formula:
\[ \sin(a + b) = \sin a \cos b + \cos a \sin b \]
we can set \( a = n \pi \) and \( b = (-1)^{n} \frac{\pi}{4} \).

### Step 3: Calculate \( \sin(n \pi) \) and \( \cos(n \pi) \)
- For any integer \( n \):
\[ \sin(n \pi) = 0 \]
\[ \cos(n \pi) = (-1)^{n} \]

### Step 4: Evaluate \( \sin\left((-1)^{n} \frac{\pi}{4}\right) \) and \( \cos\left((-1)^{n} \frac{\pi}{4}\right) \)
- When \( n \) is even, \( (-1)^{n} = 1 \):
\[ \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}, \quad \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]
- When \( n \) is odd, \( (-1)^{n} = -1 \):
\[ \sin\left(-\frac{\pi}{4}\right) = -\frac{1}{\sqrt{2}}, \quad \cos\left(-\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]

### Step 5: Substitute back into the sine addition formula
- For even \( n \):
\[ \sin\left(n \pi + \frac{\pi}{4}\right) = \sin(n \pi) \cos\left(\frac{\pi}{4}\right) + \cos(n \pi) \sin\left(\frac{\pi}{4}\right) = 0 \cdot \frac{1}{\sqrt{2}} + (-1)^{n} \cdot \frac{1}{\sqrt{2}} = \frac{1}{\sqrt{2}} \]

- For odd \( n \):
\[ \sin\left(n \pi - \frac{\pi}{4}\right) = \sin(n \pi) \cos\left(-\frac{\pi}{4}\right) + \cos(n \pi) \sin\left(-\frac{\pi}{4}\right) = 0 \cdot \frac{1}{\sqrt{2}} + (-1)^{n} \cdot \left(-\frac{1}{\sqrt{2}}\right) = \frac{1}{\sqrt{2}} \]

### Final Result
In both cases, whether \( n \) is even or odd, we find that:
\[ \sin\left(n \pi + (-1)^{n} \frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} \]

### Conclusion
Thus, the final answer is:
\[ \boxed{\frac{1}{\sqrt{2}}} \]