What is the value of the following functions (using principal value)
`tan^(-1)("tan"(5pi)/(6))`

Answer

To find the value of the function \( \tan^{-1}(\tan(5\pi/6)) \) using the principal value, we will follow these steps:

### Step 1: Find \( \tan(5\pi/6) \)

We start by recognizing that \( 5\pi/6 \) can be rewritten using the identity for tangent:
\[ \tan(5\pi/6) = \tan(\pi - \pi/6) \]
Using the property of tangent in the second quadrant, where \( \tan(\pi - x) = -\tan(x) \), we have:
\[ \tan(5\pi/6) = -\tan(\pi/6) \]

### Step 2: Calculate \( \tan(\pi/6) \)

The value of \( \tan(\pi/6) \) is known:
\[ \tan(\pi/6) = \frac{1}{\sqrt{3}} \]

### Step 3: Substitute back to find \( \tan(5\pi/6) \)

Now substituting this value back, we get:
\[ \tan(5\pi/6) = -\frac{1}{\sqrt{3}} \]

### Step 4: Find \( \tan^{-1}(-\frac{1}{\sqrt{3}}) \)

Next, we need to find \( \tan^{-1}(-\frac{1}{\sqrt{3}}) \). The angle whose tangent is \( -\frac{1}{\sqrt{3}} \) corresponds to \( -\frac{\pi}{6} \) because:
\[ \tan(-\frac{\pi}{6}) = -\tan(\frac{\pi}{6}) = -\frac{1}{\sqrt{3}} \]

### Step 5: Conclusion

Thus, the value of \( \tan^{-1}(\tan(5\pi/6)) \) using the principal value is:
\[ \tan^{-1}(\tan(5\pi/6)) = -\frac{\pi}{6} \]

### Final Answer:
\[ -\frac{\pi}{6} \]

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