Write the general solution of sin`(x+(pi)/(12))=0`

Answer

To find the general solution of the equation \( \sin\left(x + \frac{\pi}{12}\right) = 0 \), we can follow these steps:

### Step 1: Understand when the sine function is zero
The sine function is equal to zero at integer multiples of \( \pi \). This means that:
\[ \sin \theta = 0 \quad \text{when} \quad \theta = n\pi, \quad n \in \mathbb{Z} \]

### Step 2: Set up the equation
From the given equation, we have:
\[ \sin\left(x + \frac{\pi}{12}\right) = 0 \]
This implies:
\[ x + \frac{\pi}{12} = n\pi \]

### Step 3: Solve for \( x \)
Now, we can isolate \( x \) by subtracting \( \frac{\pi}{12} \) from both sides:
\[ x = n\pi - \frac{\pi}{12} \]

### Step 4: Simplify the expression
To express this in a more standard form, we can combine the terms:
\[ x = n\pi - \frac{\pi}{12} = \frac{12n\pi}{12} - \frac{\pi}{12} = \frac{(12n - 1)\pi}{12} \]

### Final General Solution
Thus, the general solution of the equation \( \sin\left(x + \frac{\pi}{12}\right) = 0 \) is:
\[ x = \frac{(12n - 1)\pi}{12}, \quad n \in \mathbb{Z} \]

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