If `sin theta + cos theta = sqrt2`, then find the value of `theta`.

Answer

To solve the equation \( \sin \theta + \cos \theta = \sqrt{2} \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \sin \theta + \cos \theta = \sqrt{2} \]

### Step 2: Divide both sides by \(\sqrt{2}\)
To simplify the equation, we divide both sides by \(\sqrt{2}\):
\[ \frac{\sin \theta}{\sqrt{2}} + \frac{\cos \theta}{\sqrt{2}} = 1 \]

### Step 3: Recognize trigonometric identities
Notice that:
\[ \frac{\sin \theta}{\sqrt{2}} = \sin \theta \cdot \frac{1}{\sqrt{2}} = \sin \theta \cdot \sin 45^\circ \]
and
\[ \frac{\cos \theta}{\sqrt{2}} = \cos \theta \cdot \frac{1}{\sqrt{2}} = \cos \theta \cdot \cos 45^\circ \]
Thus, we can rewrite the equation as:
\[ \sin \theta \cdot \sin 45^\circ + \cos \theta \cdot \cos 45^\circ = 1 \]

### Step 4: Apply the sine addition formula
Using the sine addition formula, we have:
\[ \sin(\theta + 45^\circ) = 1 \]

### Step 5: Solve for \(\theta + 45^\circ\)
The sine function equals 1 at:
\[ \theta + 45^\circ = 90^\circ + n \cdot 360^\circ \quad \text{(where \(n\) is any integer)} \]

### Step 6: Isolate \(\theta\)
Now, we isolate \(\theta\):
\[ \theta = 90^\circ - 45^\circ + n \cdot 360^\circ \]
\[ \theta = 45^\circ + n \cdot 360^\circ \]

### Step 7: Find the principal value of \(\theta\)
The principal value of \(\theta\) is:
\[ \theta = 45^\circ \]

### Final Answer
Thus, the value of \(\theta\) is:
\[ \theta = 45^\circ \]

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