If `cos3x+sin(2x-(7pi)/6)=-2` , then show that x is of the form π / 3(6m+1) .

Explanation

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

### Step 1: Analyze the Range of the Functions
The cosine function, \( \cos(3x) \), has a range of \([-1, 1]\), and the sine function, \( \sin\left(2x - \frac{7\pi}{6}\right)\), also has a range of \([-1, 1]\). Therefore, the maximum value of \( \cos(3x) + \sin\left(2x - \frac{7\pi}{6}\right) \) is \( 1 + 1 = 2\) and the minimum value is \(-1 - 1 = -2\).

### Step 2: Set Up the Equation
Given that the left-hand side equals \(-2\), we can conclude that both \( \cos(3x) \) and \( \sin\left(2x - \frac{7\pi}{6}\right) \) must attain their minimum values simultaneously:
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