The value of `tan^(-1)(tan(7pi)/6)` is

Options

A. `(7pi)/6`
B. `(5pi)/3`
C. `pi/6`
D. none of these

Answer

To find the value of \( \tan^{-1}(\tan(7\pi/6)) \), we can follow these steps:

### Step 1: Understand the function
The function \( \tan^{-1}(x) \) is the inverse of the tangent function. The output of \( \tan^{-1}(x) \) is restricted to the range \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \).

### Step 2: Calculate \( \tan(7\pi/6) \)
First, we need to find \( \tan(7\pi/6) \). The angle \( 7\pi/6 \) can be expressed in degrees:
\[ 7\pi/6 = 7 \times \frac{180}{\pi} = 210^\circ \]
The tangent function has a periodicity of \( \pi \), so we can find the equivalent angle within the range of \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \).

### Step 3: Find the reference angle
The reference angle for \( 210^\circ \) is:
\[ 210^\circ - 180^\circ = 30^\circ \]
Since \( 210^\circ \) is in the third quadrant, where tangent is positive, we have:
\[ \tan(210^\circ) = \tan(30^\circ) = \frac{1}{\sqrt{3}} \]

### Step 4: Use the periodicity of tangent
Since \( \tan(7\pi/6) = \tan(30^\circ) \), we can express this as:
\[ \tan(7\pi/6) = \tan\left(\pi + \frac{\pi}{6}\right) = \tan\left(\frac{\pi}{6}\right) \]
Thus,
\[ \tan(7\pi/6) = \frac{1}{\sqrt{3}} \]

### Step 5: Find \( \tan^{-1}(\tan(7\pi/6)) \)
Now we need to find \( \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) \). The angle whose tangent is \( \frac{1}{\sqrt{3}} \) is:
\[ \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6} \]

### Step 6: Conclusion
Since \( \frac{\pi}{6} \) is within the range of \( \tan^{-1} \), we conclude that:
\[ \tan^{-1}(\tan(7\pi/6)) = \frac{\pi}{6} \]

### Final Answer
Thus, the value of \( \tan^{-1}(\tan(7\pi/6)) \) is \( \frac{\pi}{6} \).

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