If `cos 3x+sin (2x-(7pi)/6)=-2`, then x is equal to `(k in Z)`

Options

A. `pi/3 (6k+1)`
B. `pi/3 (6k-1)`
C. `pi/3 (2k+1)`
D. none of these

Explanation

To solve the equation \( \cos(3x) + \sin\left(2x - \frac{7\pi}{6}\right) = -2 \), we can follow these steps:

### Step 1: Analyze the Range of Cosine and Sine Functions
The cosine and sine functions have a range of \([-1, 1]\). Therefore, the minimum value of \( \cos(3x) + \sin\left(2x - \frac{7\pi}{6}\right) \) can be calculated as:
\[ \cos(3x) + \sin\left(2x - \frac{7\pi}{6}\right) \geq -1 - 1 = -2 \]
This means the equation can only hold true if both \( \cos(3x) = -1 \) and \( \sin\left(2x - \frac{7\pi}{6}\right) = -1 \).

### Step 2: Solve for \( \cos(3x) = -1 \)
The cosine function equals -1 at odd multiples of \( \pi \):
\[ 3x = (2n + 1)\pi \quad \text{for } n \in \mathbb{Z} \]
Dividing both sides by 3 gives:
\[ x = \frac{(2n + 1)\pi}{3} \]

### Step 3: Solve for \( \sin\left(2x - \frac{7\pi}{6}\right) = -1 \)
The sine function equals -1 at odd multiples of \( \frac{3\pi}{2} \):
\[ 2x - \frac{7\pi}{6} = \frac{3\pi}{2} + 2m\pi \quad \text{for } m \in \mathbb{Z} \]
Rearranging gives:
\[ 2x = \frac{3\pi}{2} + 2m\pi + \frac{7\pi}{6} \]
To combine the fractions, we need a common denominator (which is 6):
\[ 2x = \frac{9\pi}{6} + \frac{12m\pi}{6} + \frac{7\pi}{6} = \frac{(9 + 12m + 7)\pi}{6} = \frac{(16 + 12m)\pi}{6} \]
Dividing both sides by 2 gives:
\[ x = \frac{(16 + 12m)\pi}{12} = \frac{(4 + 3m)\pi}{3} \]

### Step 4: Set the Two Expressions for \( x \) Equal
Now we have two expressions for \( x \):
1. \( x = \frac{(2n + 1)\pi}{3} \)
2. \( x = \frac{(4 + 3m)\pi}{3} \)

Equating them gives:
\[ \frac{(2n + 1)\pi}{3} = \frac{(4 + 3m)\pi}{3} \]
Removing \( \frac{\pi}{3} \) from both sides leads to:
\[ 2n + 1 = 4 + 3m \]
Rearranging this gives:
\[ 2n - 3m = 3 \]

### Step 5: General Solution
The general solution can be expressed in terms of integers \( k \) (where \( k \) is any integer):
\[ x = \frac{(2k + 1)\pi}{3} \quad \text{or} \quad x = \frac{(4 + 3k)\pi}{3} \]

### Final Answer
Thus, the solution for \( x \) can be expressed as:
\[ x = \frac{(2k + 1)\pi}{3} \quad \text{where } k \in \mathbb{Z} \]