find the period of sin`(x/2)-cos(x/3)` is

Explanation

To find the period of the function \( f(x) = \sin\left(\frac{x}{2}\right) - \cos\left(\frac{x}{3}\right) \), we need to determine the individual periods of the sine and cosine components and then find the least common multiple (LCM) of these periods.

### Step 1: Determine the period of \( \sin\left(\frac{x}{2}\right) \)

The standard period of the sine function \( \sin(x) \) is \( 2\pi \). When the argument of the sine function is modified to \( \frac{x}{2} \), the period changes according to the formula:

\[ \text{Period} = \frac{2\pi}{k} \]

where \( k \) is the coefficient of \( x \) in the argument. Here, \( k = \frac{1}{2} \), so:

\[ \text{Period of } \sin\left(\frac{x}{2}\right) = \frac{2\pi}{\frac{1}{2}} = 4\pi \]

### Step 2: Determine the period of \( \cos\left(\frac{x}{3}\right) \)

Similarly, the standard period of the cosine function \( \cos(x) \) is also \( 2\pi \). For the argument \( \frac{x}{3} \), we apply the same formula:

\[ \text{Period of } \cos\left(\frac{x}{3}\right) = \frac{2\pi}{\frac{1}{3}} = 6\pi \]

### Step 3: Find the LCM of the periods

Now, we have the periods of both functions:
- Period of \( \sin\left(\frac{x}{2}\right) = 4\pi \)
- Period of \( \cos\left(\frac{x}{3}\right) = 6\pi \)

To find the period of the combined function \( f(x) \), we need to calculate the least common multiple (LCM) of \( 4\pi \) and \( 6\pi \).

The LCM of the coefficients \( 4 \) and \( 6 \) is \( 12 \). Therefore:

\[ \text{LCM}(4\pi, 6\pi) = 12\pi \]

### Conclusion

Thus, the period of the function \( f(x) = \sin\left(\frac{x}{2}\right) - \cos\left(\frac{x}{3}\right) \) is:

\[ \text{Period} = 12\pi \]

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