Principal value of `tan^(-1)[tan((3pi)/4)]` is

Options

A. `pi/4`
B. `(3pi)/4`
C. `-pi/4`
D. `-(3pi)/4`

Answer

To find the principal value of \( \tan^{-1}(\tan(3\pi/4)) \), we will follow these steps:

### Step 1: Understand the range of the inverse tangent function
The function \( \tan^{-1}(x) \) has a range of \( \left(-\frac{\pi}{2}, \frac{\pi}{2}\right) \). This means that the output of \( \tan^{-1} \) will always lie within this interval.

**Hint:** Remember that the range of the inverse tangent function is limited to the first and fourth quadrants.

### Step 2: Evaluate \( \tan(3\pi/4) \)
We know that \( 3\pi/4 \) is in the second quadrant. The tangent function is negative in the second quadrant. We can express \( 3\pi/4 \) as:
\[ 3\pi/4 = \pi - \pi/4 \]
Using the identity \( \tan(\pi - \theta) = -\tan(\theta) \), we have:
\[ \tan(3\pi/4) = \tan(\pi - \pi/4) = -\tan(\pi/4) \]
Since \( \tan(\pi/4) = 1 \), it follows that:
\[ \tan(3\pi/4) = -1 \]

**Hint:** Use the identity for tangent in different quadrants to find the value.

### Step 3: Substitute back into the inverse tangent function
Now we substitute back into our original expression:
\[ \tan^{-1}(\tan(3\pi/4)) = \tan^{-1}(-1) \]

**Hint:** Recall that \( \tan^{-1}(-x) = -\tan^{-1}(x) \).

### Step 4: Find \( \tan^{-1}(-1) \)
We know that:
\[ \tan^{-1}(-1) = -\frac{\pi}{4} \]

**Hint:** Recognize that the angle whose tangent is -1 is \( -\frac{\pi}{4} \).

### Conclusion
Thus, the principal value of \( \tan^{-1}(\tan(3\pi/4)) \) is:
\[ -\frac{\pi}{4} \]

**Final Answer:**
\[ -\frac{\pi}{4} \]