Find the period of the following, if exists:
f(x) = tan3x + sin(x/3)

Answer

To find the period of the function \( f(x) = \tan(3x) + \sin\left(\frac{x}{3}\right) \), we need to determine the periods of each component of the function separately and then find the least common multiple (LCM) of these periods.

### Step 1: Determine the period of \( \tan(3x) \)

The period of the tangent function \( \tan(x) \) is \( \pi \). When the function is transformed to \( \tan(kx) \), the period changes to \( \frac{\pi}{k} \).

Here, \( k = 3 \), so the period of \( \tan(3x) \) is:
\[ \text{Period of } \tan(3x) = \frac{\pi}{3} \]

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

The period of the sine function \( \sin(x) \) is \( 2\pi \). When the function is transformed to \( \sin\left(\frac{x}{k}\right) \), the period changes to \( 2\pi k \).

Here, \( k = \frac{1}{3} \), so the period of \( \sin\left(\frac{x}{3}\right) \) is:
\[ \text{Period of } \sin\left(\frac{x}{3}\right) = 2\pi \cdot 3 = 6\pi \]

### Step 3: Find the least common multiple (LCM) of the two periods

Now we have:
- Period of \( \tan(3x) = \frac{\pi}{3} \)
- Period of \( \sin\left(\frac{x}{3}\right) = 6\pi \)

To find the LCM, we first express both periods with a common denominator:
- \( \frac{\pi}{3} = \frac{\pi}{3} \)
- \( 6\pi = \frac{18\pi}{3} \)

Now, we can find the LCM of the numerators \( \pi \) and \( 18\pi \):
- The LCM of \( \pi \) and \( 18\pi \) is \( 18\pi \).

Since the denominators are the same (both are 3), the LCM of the two periods is:
\[ \text{LCM} = \frac{18\pi}{3} = 6\pi \]

### Conclusion

The period of the function \( f(x) = \tan(3x) + \sin\left(\frac{x}{3}\right) \) is:
\[ \text{Period of } f(x) = 6\pi \]