Find period of following functions (if existsI)
`f(x)=sin3x+tan7x`

Answer

To find the period of the function \( f(x) = \sin(3x) + \tan(7x) \), we will follow these steps:

### Step 1: Determine the period of \( \sin(3x) \)
The period of the sine function \( \sin(x) \) is \( 2\pi \). For \( \sin(mx) \), the period is given by:

\[ T_1 = \frac{2\pi}{m} \]

In our case, \( m = 3 \):

\[ T_1 = \frac{2\pi}{3} \]

### Step 2: Determine the period of \( \tan(7x) \)
The period of the tangent function \( \tan(x) \) is \( \pi \). For \( \tan(mx) \), the period is given by:

\[ T_2 = \frac{\pi}{m} \]

In our case, \( m = 7 \):

\[ T_2 = \frac{\pi}{7} \]

### Step 3: Find the least common multiple (LCM) of the periods
To find the period of the sum of the two functions, we need to find the LCM of \( T_1 \) and \( T_2 \):

\[ T_1 = \frac{2\pi}{3}, \quad T_2 = \frac{\pi}{7} \]

The LCM of two fractions \( \frac{a}{b} \) and \( \frac{c}{d} \) is given by:

\[ \text{LCM}\left(\frac{a}{b}, \frac{c}{d}\right) = \frac{\text{LCM}(a, c)}{\text{GCD}(b, d)} \]

Here, \( a = 2\pi \), \( b = 3 \), \( c = \pi \), and \( d = 7 \).

### Step 4: Calculate LCM of the numerators
\[ \text{LCM}(2\pi, \pi) = 2\pi \]

### Step 5: Calculate GCD of the denominators
Since 3 and 7 are coprime:

\[ \text{GCD}(3, 7) = 1 \]

### Step 6: Calculate the overall LCM
Now, we can find the LCM of the two periods:

\[ \text{LCM}\left(\frac{2\pi}{3}, \frac{\pi}{7}\right) = \frac{2\pi}{1} = 2\pi \]

### Conclusion
Thus, the period of the function \( f(x) = \sin(3x) + \tan(7x) \) is:

\[ \boxed{2\pi} \]