Maximum value of `sin x+cos x` is:

Options

A. 1
B. 2
C. `sqrt2`
D. `(1)/(sqrt2)`

Answer

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

### Step 1: Rewrite the expression
We start with the expression:
\[ y = \sin x + \cos x \]

### Step 2: Use the identity for maximum value
We know that the maximum value of \( A \sin x + B \cos x \) is given by:
\[ \sqrt{A^2 + B^2} \]
In our case, \( A = 1 \) and \( B = 1 \).

### Step 3: Calculate \( A^2 + B^2 \)
Now we compute:
\[ A^2 + B^2 = 1^2 + 1^2 = 1 + 1 = 2 \]

### Step 4: Take the square root
Next, we take the square root:
\[ \sqrt{A^2 + B^2} = \sqrt{2} \]

### Step 5: Conclusion
Thus, the maximum value of \( \sin x + \cos x \) is:
\[ \sqrt{2} \]