The most general solution of the equation `sqrt3cosx +sinx = sqrt2` is :

Options

A. `2npi +(5pi)/(12)`
B. `2npi pm pi/4 + pi/6`
C. `2npi - pi/(12)`
D. None of these

Answer

To find the most general solution of the equation \( \sqrt{3} \cos x + \sin x = \sqrt{2} \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the given 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 values
Next, we recognize that:
\[ \frac{\sqrt{3}}{2} = \cos \frac{\pi}{6}, \quad \frac{1}{2} = \sin \frac{\pi}{6}, \quad \text{and} \quad \frac{\sqrt{2}}{2} = \cos \frac{\pi}{4} \]
Thus, we can rewrite the equation as:
\[ \cos \frac{\pi}{6} \cos x + \sin \frac{\pi}{6} \sin x = \cos \frac{\pi}{4} \]

### Step 4: Use the cosine addition formula
Using the cosine addition formula \( \cos A \cos B + \sin A \sin B = \cos(A - B) \), we rewrite the left-hand side:
\[ \cos\left(x - \frac{\pi}{6}\right) = \cos \frac{\pi}{4} \]

### Step 5: Solve for \( x \)
From the equation \( \cos A = \cos B \), we know that:
\[ A = 2n\pi \pm B \]
Thus, we have:
\[ x - \frac{\pi}{6} = 2n\pi \pm \frac{\pi}{4} \]

### Step 6: Isolate \( x \)
Now, we isolate \( x \):
\[ x = 2n\pi + \frac{\pi}{4} + \frac{\pi}{6} \quad \text{or} \quad x = 2n\pi - \frac{\pi}{4} + \frac{\pi}{6} \]

### Step 7: Simplify the expressions
To simplify further, we need a common denominator for the fractions:
1. For \( \frac{\pi}{4} + \frac{\pi}{6} \):
- The common denominator is 12.
- Thus, \( \frac{\pi}{4} = \frac{3\pi}{12} \) and \( \frac{\pi}{6} = \frac{2\pi}{12} \).
- Therefore, \( \frac{\pi}{4} + \frac{\pi}{6} = \frac{5\pi}{12} \).

2. For \( -\frac{\pi}{4} + \frac{\pi}{6} \):
- Thus, \( -\frac{\pi}{4} = -\frac{3\pi}{12} \).
- Therefore, \( -\frac{\pi}{4} + \frac{\pi}{6} = -\frac{3\pi}{12} + \frac{2\pi}{12} = -\frac{\pi}{12} \).

### Final General Solution
Thus, the most general solutions are:
\[ x = 2n\pi + \frac{5\pi}{12} \quad \text{and} \quad x = 2n\pi - \frac{\pi}{12} \]

### Summary
The most general solution of the equation \( \sqrt{3} \cos x + \sin x = \sqrt{2} \) is:
\[ x = 2n\pi + \frac{5\pi}{12} \quad \text{or} \quad x = 2n\pi - \frac{\pi}{12} \]