Find the maximum value of `1+sin(pi/4+theta) + 2cos(pi/4-theta)`.

Options

A. 1
B. 2
C. 3
D. 4

Explanation

To find the maximum value of the expression \(1 + \sin\left(\frac{\pi}{4} + \theta\right) + 2\cos\left(\frac{\pi}{4} - \theta\right)\), we can follow these steps:

### Step 1: Rewrite the expression using trigonometric identities
Using the sine and cosine addition formulas:
\[ \sin(a + b) = \sin a \cos b + \cos a \sin b \]
\[ \cos(a - b) = \cos a \cos b + \sin a \sin b \]
we can rewrite the expression as:
\[ 1 + \left(\sin\left(\frac{\pi}{4}\right)\cos(\theta) + \cos\left(\frac{\pi}{4}\right)\sin(\theta)\right) + 2\left(\cos\left(\frac{\pi}{4}\right)\cos(\theta) + \sin\left(\frac{\pi}{4}\right)\sin(\theta)\right) \]

### Step 2: Substitute the values of \(\sin\left(\frac{\pi}{4}\right)\) and \(\cos\left(\frac{\pi}{4}\right)\)
Since \(\sin\left(\frac{\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}\), we substitute these values:
\[ 1 + \left(\frac{1}{\sqrt{2}}\cos(\theta) + \frac{1}{\sqrt{2}}\sin(\theta)\right) + 2\left(\frac{1}{\sqrt{2}}\cos(\theta) + \frac{1}{\sqrt{2}}\sin(\theta)\right) \]

### Step 3: Simplify the expression
Combining the terms, we have:
\[ 1 + \frac{1}{\sqrt{2}}\cos(\theta) + \frac{1}{\sqrt{2}}\sin(\theta) + \sqrt{2}\cos(\theta) + \sqrt{2}\sin(\theta) \]
This can be rewritten as:
\[ 1 + \left(\frac{1}{\sqrt{2}} + \sqrt{2}\right)\cos(\theta) + \left(\frac{1}{\sqrt{2}} + \sqrt{2}\right)\sin(\theta) \]

### Step 4: Factor out the common terms
Let \(k = \frac{1}{\sqrt{2}} + \sqrt{2}\):
\[ 1 + k(\cos(\theta) + \sin(\theta)) \]

### Step 5: Find the maximum value of \(\cos(\theta) + \sin(\theta)\)
The maximum value of \(\cos(\theta) + \sin(\theta)\) can be found using the identity:
\[ \cos(\theta) + \sin(\theta) = \sqrt{2}\sin\left(\theta + \frac{\pi}{4}\right) \]
The maximum value of \(\sin\) is 1, so:
\[ \max(\cos(\theta) + \sin(\theta)) = \sqrt{2} \]

### Step 6: Substitute back to find the maximum value of the expression
Thus, the maximum value of the original expression is:
\[ 1 + k\sqrt{2} = 1 + \left(\frac{1}{\sqrt{2}} + \sqrt{2}\right)\sqrt{2} \]
Calculating \(k\sqrt{2}\):
\[ k\sqrt{2} = \left(\frac{1}{\sqrt{2}} + \sqrt{2}\right)\sqrt{2} = 1 + 2 = 3 \]
Therefore, the maximum value of the expression is:
\[ 1 + 3 = 4 \]

### Final Answer
The maximum value of \(1 + \sin\left(\frac{\pi}{4} + \theta\right) + 2\cos\left(\frac{\pi}{4} - \theta\right)\) is **4**.