Write the value of tan`((19pi)/(3))`

Answer

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

### Step 1: Simplify the angle
First, we need to simplify the angle \( \frac{19\pi}{3} \) by reducing it within the range of \( [0, 2\pi) \). We do this by subtracting multiples of \( 2\pi \).

To find how many times \( 2\pi \) fits into \( \frac{19\pi}{3} \):
\[ 2\pi = \frac{6\pi}{3} \]
Now, we divide \( \frac{19\pi}{3} \) by \( 2\pi \):
\[ \frac{19\pi/3}{2\pi} = \frac{19}{6} \]
The integer part of \( \frac{19}{6} \) is 3, meaning we can subtract \( 3 \times 2\pi \) from \( \frac{19\pi}{3} \):
\[ \frac{19\pi}{3} - 3 \times 2\pi = \frac{19\pi}{3} - \frac{18\pi}{3} = \frac{\pi}{3} \]

### Step 2: Use the periodic property of the tangent function
The tangent function has a period of \( \pi \):
\[ \tan(\theta + n\pi) = \tan(\theta) \]
Thus, we have:
\[ \tan\left(\frac{19\pi}{3}\right) = \tan\left(\frac{\pi}{3}\right) \]

### Step 3: Find the value of \( \tan\left(\frac{\pi}{3}\right) \)
Now, we can find the value of \( \tan\left(\frac{\pi}{3}\right) \):
\[ \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \]

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

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