Maximum value of `sin (x + (pi)/(6)) + cos (x + (pi)/(6))` is

Options

A. `(pi)/12`
B. `(pi)/6`
C. `(pi)/3`
D. `(pi)/2`

Answer

To find the maximum value of the expression \( \sin\left(x + \frac{\pi}{6}\right) + \cos\left(x + \frac{\pi}{6}\right) \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ y = \sin\left(x + \frac{\pi}{6}\right) + \cos\left(x + \frac{\pi}{6}\right) \]

### Step 2: Use the identity for sine and cosine
We can factor out \( \sqrt{2} \) by multiplying and dividing by \( \sqrt{2} \):
\[ y = \frac{1}{\sqrt{2}} \sqrt{2} \left( \sin\left(x + \frac{\pi}{6}\right) + \cos\left(x + \frac{\pi}{6}\right) \right) \]

### Step 3: Recognize the sine addition formula
Using the sine addition formula, we can rewrite the expression:
\[ y = \frac{1}{\sqrt{2}} \left( \sin\left(x + \frac{\pi}{6}\right) + \cos\left(x + \frac{\pi}{6}\right) \right) = \frac{1}{\sqrt{2}} \left( \sin\left(x + \frac{\pi}{6} + \frac{\pi}{4}\right) \right) \]
This simplifies to:
\[ y = \frac{1}{\sqrt{2}} \sin\left(x + \frac{\pi}{6} + \frac{\pi}{4}\right) \]

### Step 4: Find the maximum value
The maximum value of \( \sin \) function is 1. Therefore, the maximum value of \( y \) occurs when:
\[ \sin\left(x + \frac{\pi}{6} + \frac{\pi}{4}\right) = 1 \]
This leads to:
\[ y_{\text{max}} = \frac{1}{\sqrt{2}} \cdot 1 = \frac{1}{\sqrt{2}} \cdot \sqrt{2} = 1 \]

### Step 5: Final value
Thus, the maximum value of the original expression \( \sin\left(x + \frac{\pi}{6}\right) + \cos\left(x + \frac{\pi}{6}\right) \) is:
\[ \sqrt{2} \]

### Conclusion
The maximum value of \( \sin\left(x + \frac{\pi}{6}\right) + \cos\left(x + \frac{\pi}{6}\right) \) is \( \sqrt{2} \).