Question

The value of \( \tan^{-1} \left( \tan \frac{7\pi}{6} \right) \text{ is:} \)

• A. \( \frac{\pi}{6} \)
• B. \( \frac{\pi}{3} \)
• C. \( \frac{5\pi}{6} \)
• D. \( \frac{7\pi}{6} \)

Hint:

When evaluating inverse trigonometric functions, ensure the angle is within the principal range of the function. If it's not, adjust the angle accordingly by adding or subtracting multiples of \( \pi \).

Answer 1

The Correct Option is A

Step 1: Recognize the function and angle.
We are asked to evaluate the value of \( \tan^{-1} \left( \tan \frac{7\pi}{6} \right) \). Here, the function is the inverse tangent (also known as arctan), and we need to compute the angle corresponding to the given expression.

Step 2: Simplify the expression.
Recall that \( \tan^{-1} (\tan x) = x \) only when \( x \) is within the principal range of the arctan function, which is \( -\frac{\pi}{2} \leq x \leq \frac{\pi}{2} \).

Step 3: Adjust the angle.
The angle \( \frac{7\pi}{6} \) is outside this range. We need to find an equivalent angle within the range \( -\frac{\pi}{2} \leq x \leq \frac{\pi}{2} \).
Since \( \frac{7\pi}{6} \) is greater than \( \pi \), it lies in the third quadrant. We can subtract \( \pi \) from \( \frac{7\pi}{6} \) to bring it into the desired range:
\[ \frac{7\pi}{6} - \pi = \frac{7\pi}{6} - \frac{6\pi}{6} = \frac{\pi}{6}. \]

Step 4: Use the inverse tangent identity.
Now we compute:
\[ \tan^{-1} \left( \tan \frac{7\pi}{6} \right) = \tan^{-1} \left( \tan \frac{\pi}{6} \right). \]
Since \( \frac{\pi}{6} \) is within the principal range, we can directly conclude that:
\[ \tan^{-1} \left( \tan \frac{\pi}{6} \right) = \frac{\pi}{6}. \]

Step 5: Conclusion.
Thus, the value of \( \tan^{-1} \left( \tan \frac{7\pi}{6} \right) \) is \( \frac{\pi}{6} \).
Final Answer: \[ \boxed{\frac{\pi}{6}}. \]