Solve the equation `sqrt3cos x + sin x = sqrt2 `.

Answer

To solve the equation \( \sqrt{3} \cos x + \sin x = \sqrt{2} \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the original equation:
\[ \sqrt{3} \cos x + \sin x = \sqrt{2} \]

### Step 2: Divide both sides by 2
To simplify the equation, we can divide both sides by 2:
\[ \frac{\sqrt{3}}{2} \cos x + \frac{1}{2} \sin x = \frac{\sqrt{2}}{2} \]

### Step 3: Recognize trigonometric identities
We know that:
\[ \frac{\sqrt{3}}{2} = \sin\left(\frac{\pi}{3}\right) \quad \text{and} \quad \frac{1}{2} = \cos\left(\frac{\pi}{3}\right) \]
Thus, we can rewrite the left-hand side using the sine addition formula:
\[ \sin\left(\frac{\pi}{3}\right) \cos x + \cos\left(\frac{\pi}{3}\right) \sin x = \sin\left(x + \frac{\pi}{3}\right) \]

### Step 4: Rewrite the equation using sine
Now we can rewrite the equation as:
\[ \sin\left(x + \frac{\pi}{3}\right) = \frac{\sqrt{2}}{2} \]

### Step 5: Find general solutions for sine
The sine function equals \( \frac{\sqrt{2}}{2} \) at:
\[ x + \frac{\pi}{3} = n\pi + \frac{\pi}{4} \quad \text{or} \quad x + \frac{\pi}{3} = n\pi + \frac{3\pi}{4} \]
where \( n \) is any integer.

### Step 6: Solve for \( x \)
1. From the first equation:
\[ x + \frac{\pi}{3} = n\pi + \frac{\pi}{4} \]
Rearranging gives:
\[ x = n\pi + \frac{\pi}{4} - \frac{\pi}{3} \]
To combine the fractions:
\[ x = n\pi + \left(\frac{3\pi}{12} - \frac{4\pi}{12}\right) = n\pi - \frac{\pi}{12} \]

2. From the second equation:
\[ x + \frac{\pi}{3} = n\pi + \frac{3\pi}{4} \]
Rearranging gives:
\[ x = n\pi + \frac{3\pi}{4} - \frac{\pi}{3} \]
To combine the fractions:
\[ x = n\pi + \left(\frac{9\pi}{12} - \frac{4\pi}{12}\right) = n\pi + \frac{5\pi}{12} \]

### Final Solutions
Thus, the general solutions for \( x \) are:
\[ x = n\pi - \frac{\pi}{12} \quad \text{and} \quad x = n\pi + \frac{5\pi}{12} \]
where \( n \) is any integer.

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