Find the minimum value of sin x + cos x.

Answer

To find the minimum value of \( \sin x + \cos x \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ y = \sin x + \cos x \]
We can factor out \( \sqrt{2} \) to help us rewrite the expression.

### Step 2: Factor out \( \sqrt{2} \)
We can express \( \sin x + \cos x \) as:
\[ y = \sqrt{2} \left( \frac{1}{\sqrt{2}} \sin x + \frac{1}{\sqrt{2}} \cos x \right) \]
Here, we have multiplied and divided by \( \sqrt{2} \).

### Step 3: Recognize the angle
Notice that \( \frac{1}{\sqrt{2}} = \sin\left(\frac{\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right) \). Thus, we can rewrite the expression as:
\[ y = \sqrt{2} \left( \sin\left(\frac{\pi}{4}\right) \sin x + \cos\left(\frac{\pi}{4}\right) \cos x \right) \]
This is the sine addition formula:
\[ y = \sqrt{2} \sin\left(x + \frac{\pi}{4}\right) \]

### Step 4: Determine the range of the sine function
The sine function \( \sin(x + \frac{\pi}{4}) \) oscillates between -1 and 1. Therefore, the minimum value of \( y \) occurs when:
\[ \sin\left(x + \frac{\pi}{4}\right) = -1 \]

### Step 5: Calculate the minimum value
Substituting this back into our expression for \( y \):
\[ y_{\text{min}} = \sqrt{2} \cdot (-1) = -\sqrt{2} \]

### Conclusion
Thus, the minimum value of \( \sin x + \cos x \) is:
\[ \boxed{-\sqrt{2}} \]