Solve the equation `sinx + sqrt(3)cosx=sqrt(2)`

Options

A. `x = npi +(-1)^npi/4-pi/3, n int l`
B. `x =2npi - (5pi)/12, n int I`
C. `x = npi + (5pi)/12, n int I`
D. `x = 2npi+pi/12, n int I`

Answer

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

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

### Step 2: Divide by 2
Next, we divide the entire equation by 2:
\[ \frac{\sqrt{3}}{2} \cos x + \frac{1}{2} \sin x = \frac{\sqrt{2}}{2} \]

### Step 3: Recognize trigonometric values
We know that:
\[ \frac{\sqrt{3}}{2} = \cos\left(\frac{\pi}{6}\right) \quad \text{and} \quad \frac{1}{2} = \sin\left(\frac{\pi}{6}\right) \]
Thus, we can rewrite the left-hand side using the cosine of a sum formula:
\[ \cos\left(\frac{\pi}{6}\right) \cos x + \sin\left(\frac{\pi}{6}\right) \sin x = \frac{\sqrt{2}}{2} \]

### Step 4: Use the cosine addition formula
Using the cosine addition formula \( \cos(a - b) = \cos a \cos b + \sin a \sin b \), we have:
\[ \cos\left(x - \frac{\pi}{6}\right) = \frac{\sqrt{2}}{2} \]

### Step 5: Solve for \( x \)
The equation \( \cos\left(x - \frac{\pi}{6}\right) = \frac{\sqrt{2}}{2} \) implies:
\[ x - \frac{\pi}{6} = 2n\pi \pm \frac{\pi}{4} \quad \text{for } n \in \mathbb{Z} \]
This gives us two equations:
1. \( x - \frac{\pi}{6} = 2n\pi + \frac{\pi}{4} \)
2. \( x - \frac{\pi}{6} = 2n\pi - \frac{\pi}{4} \)

### Step 6: Solve each equation for \( x \)
For the first equation:
\[ x = 2n\pi + \frac{\pi}{4} + \frac{\pi}{6} \]
Finding a common denominator (12):
\[ x = 2n\pi + \frac{3\pi}{12} + \frac{2\pi}{12} = 2n\pi + \frac{5\pi}{12} \]

For the second equation:
\[ x = 2n\pi - \frac{\pi}{4} + \frac{\pi}{6} \]
Again, finding a common denominator (12):
\[ x = 2n\pi - \frac{3\pi}{12} + \frac{2\pi}{12} = 2n\pi - \frac{\pi}{12} \]

### Final Solutions
Thus, the solutions are:
1. \( x = 2n\pi + \frac{5\pi}{12} \)
2. \( x = 2n\pi - \frac{\pi}{12} \)

### Summary of Solutions
The final solutions can be summarized as:
- \( x = 2n\pi + \frac{5\pi}{12} \)
- \( x = 2n\pi - \frac{\pi}{12} \)