The period of the function `f(x)=sin((2x+3)/(6pi))`, is

Options

A. `2pi`
B. `6pi`
C. `6pi^(2)`
D. none of these

Answer

To find the period of the function \( f(x) = \sin\left(\frac{2x + 3}{6\pi}\right) \), we can follow these steps:

### Step 1: Identify the function
We start with the function:
\[ f(x) = \sin\left(\frac{2x + 3}{6\pi}\right) \]

### Step 2: Rewrite the function
We can separate the terms inside the sine function:
\[ f(x) = \sin\left(\frac{2x}{6\pi} + \frac{3}{6\pi}\right) = \sin\left(\frac{x}{3\pi} + \frac{1}{2\pi}\right) \]

### Step 3: Determine the coefficient of \(x\)
In the expression \(\sin\left(\frac{x}{3\pi} + \frac{1}{2\pi}\right)\), the coefficient of \(x\) is \(\frac{1}{3\pi}\).

### Step 4: Use the formula for the period of the sine function
The period \(T\) of the sine function \(\sin(kx)\) is given by:
\[ T = \frac{2\pi}{k} \]
where \(k\) is the coefficient of \(x\).

### Step 5: Substitute the coefficient into the period formula
Here, \(k = \frac{1}{3\pi}\). Therefore, the period \(T\) is:
\[ T = \frac{2\pi}{\frac{1}{3\pi}} = 2\pi \times 3\pi = 6\pi^2 \]

### Conclusion
Thus, the period of the function \( f(x) = \sin\left(\frac{2x + 3}{6\pi}\right) \) is:
\[ \boxed{6\pi^2} \]

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