Solve `: sqrt3 sin theta - cos theta = sqrt2.`

Answer

To solve the equation \( \sqrt{3} \sin \theta - \cos \theta = \sqrt{2} \), we can follow these steps:

### Step 1: Rearrange the equation
We start with the given equation:
\[ \sqrt{3} \sin \theta - \cos \theta = \sqrt{2} \]
We can rearrange it to isolate the trigonometric terms on one side:
\[ \sqrt{3} \sin \theta - \sqrt{2} = \cos \theta \]

### Step 2: Divide by 2
Next, we can divide the entire equation by 2 to simplify:
\[ \frac{\sqrt{3}}{2} \sin \theta - \frac{1}{2} \cos \theta = \frac{\sqrt{2}}{2} \]
This simplifies to:
\[ \frac{\sqrt{3}}{2} \sin \theta - \frac{1}{2} \cos \theta = \frac{1}{\sqrt{2}} \]

### Step 3: Recognize trigonometric identities
We know that:
\[ \frac{\sqrt{3}}{2} = \cos 30^\circ \quad \text{and} \quad \frac{1}{2} = \sin 30^\circ \]
Thus, we can rewrite the left-hand side using the sine subtraction formula:
\[ \cos 30^\circ \sin \theta - \sin 30^\circ \cos \theta = \frac{1}{\sqrt{2}} \]
This can be expressed as:
\[ \sin(\theta - 30^\circ) = \frac{1}{\sqrt{2}} \]

### Step 4: Solve for angles
The equation \( \sin(\theta - 30^\circ) = \frac{1}{\sqrt{2}} \) implies:
\[ \theta - 30^\circ = n\pi + (-1)^n \frac{\pi}{4} \]
where \( n \) is any integer.

### Step 5: Solve for \( \theta \)
Now, we can solve for \( \theta \):
\[ \theta = n\pi + (-1)^n \frac{\pi}{4} + 30^\circ \]
Converting \( 30^\circ \) to radians gives us \( \frac{\pi}{6} \):
\[ \theta = n\pi + (-1)^n \frac{\pi}{4} + \frac{\pi}{6} \]

### Step 6: Combine terms
To combine the terms, we can find a common denominator:
\[ \theta = n\pi + \frac{3(-1)^n + 2}{12}\pi \]

### Final Answer
Thus, the general solution for \( \theta \) is:
\[ \theta = n\pi + \frac{3(-1)^n + 2}{12}\pi, \quad n \in \mathbb{Z} \]