Find the value of the trigonometric function tan`(19pi)/3`

Options

A. `sqrt3`
B. `sqrt4`
C. `sqrt6`
D. `sqrt2`

Explanation

To find the value of the trigonometric function \( \tan\left(\frac{19\pi}{3}\right) \), we can follow these steps:

### Step 1: Simplify the angle
First, we need to simplify \( \frac{19\pi}{3} \) by reducing it to an equivalent angle within the range of \( 0 \) to \( 2\pi \). We can do this by subtracting \( 2\pi \) (which is equivalent to \( \frac{6\pi}{3} \)) multiple times until we get an angle in the desired range.

\[ \frac{19\pi}{3} - 2\pi = \frac{19\pi}{3} - \frac{6\pi}{3} = \frac{13\pi}{3} \]

Next, we subtract \( 2\pi \) again:

\[ \frac{13\pi}{3} - 2\pi = \frac{13\pi}{3} - \frac{6\pi}{3} = \frac{7\pi}{3} \]

Now we subtract \( 2\pi \) one more time:

\[ \frac{7\pi}{3} - 2\pi = \frac{7\pi}{3} - \frac{6\pi}{3} = \frac{\pi}{3} \]

So, we have:

\[ \tan\left(\frac{19\pi}{3}\right) = \tan\left(\frac{\pi}{3}\right) \]

### Step 2: Find the value of \( \tan\left(\frac{\pi}{3}\right) \)
Now we can find the value of \( \tan\left(\frac{\pi}{3}\right) \). We know from trigonometric values that:

\[ \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \]

### Conclusion
Thus, the value of \( \tan\left(\frac{19\pi}{3}\right) \) is:

\[ \boxed{\sqrt{3}} \]