General solution of `sin((3x)/(2))=0` is

Options

A. `(npi)/(3), ninZ`
B. `(2npi)/(3), ninZ`
C. `npi, ninZ`
D. `(3npi)/(2), ninZ`

Answer

To find the general solution of the equation \( \sin\left(\frac{3x}{2}\right) = 0 \), we can follow these steps:

### Step 1: Set the argument of sine equal to zero
We start by letting \( \theta = \frac{3x}{2} \). Therefore, we need to solve:
\[ \sin(\theta) = 0 \]

### Step 2: Identify the general solutions for sine
The sine function equals zero at integer multiples of \( \pi \):
\[ \theta = n\pi \quad \text{where } n \in \mathbb{Z} \text{ (the set of all integers)} \]

### Step 3: Substitute back for \( \theta \)
Now, we substitute back \( \theta = \frac{3x}{2} \):
\[ \frac{3x}{2} = n\pi \]

### Step 4: Solve for \( x \)
To isolate \( x \), we multiply both sides by \( \frac{2}{3} \):
\[ x = \frac{2n\pi}{3} \]

### Step 5: Write the final general solution
Thus, the general solution for the original equation \( \sin\left(\frac{3x}{2}\right) = 0 \) is:
\[ x = \frac{2n\pi}{3} \quad \text{where } n \in \mathbb{Z} \]