Find the values 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 simplify the angle using the periodic property of the tangent function. Here’s a step-by-step solution:

### Step 1: Understand the periodicity of the tangent function
The tangent function has a period of \( \pi \). This means that:
\[ \tan(\theta + n\pi) = \tan(\theta) \]
for any integer \( n \).

### Step 2: Simplify the angle \( \frac{19\pi}{3} \)
To simplify \( \frac{19\pi}{3} \), we can subtract multiples of \( 2\pi \) (which is equivalent to \( \frac{6\pi}{3} \)) to bring the angle within the range of \( 0 \) to \( 2\pi \).

First, we can find how many full \( 2\pi \) (or \( \frac{6\pi}{3} \)) fit into \( \frac{19\pi}{3} \):
\[ \frac{19\pi/3}{6\pi/3} = \frac{19}{6} \approx 3.1667 \]
This means we can subtract \( 3 \times 2\pi \) from \( \frac{19\pi}{3} \).

### Step 3: Calculate the equivalent angle
Now, we subtract \( 3 \times 2\pi \):
\[ \frac{19\pi}{3} - 3 \times 2\pi = \frac{19\pi}{3} - \frac{18\pi}{3} = \frac{\pi}{3} \]

### Step 4: Find the tangent of the simplified angle
Now we can find the tangent of the simplified angle:
\[ \tan\left(\frac{19\pi}{3}\right) = \tan\left(\frac{\pi}{3}\right) \]

### Step 5: Use known values of tangent
The value of \( \tan\left(\frac{\pi}{3}\right) \) is a known trigonometric value:
\[ \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \]

### Final Answer
Thus, the value of \( \tan\left(\frac{19\pi}{3}\right) \) is:
\[ \boxed{\sqrt{3}} \]