What is the principal argument of `(-1-i),` where `i=sqrt(-1)` ?

Options

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

Explanation

To find the principal argument of the complex number \(-1 - i\), we can follow these steps:

### Step 1: Identify the complex number
The complex number given is:
\[ z = -1 - i \]
Here, the real part \(a = -1\) and the imaginary part \(b = -1\).

### Step 2: Determine the quadrant
The complex number \(-1 - i\) lies in the third quadrant because both the real part and the imaginary part are negative.

### Step 3: Calculate the argument
The argument \(\theta\) of a complex number can be calculated using the formula:
\[ \theta = \tan^{-1}\left(\frac{b}{a}\right) \]
Substituting the values of \(a\) and \(b\):
\[ \theta = \tan^{-1}\left(\frac{-1}{-1}\right) = \tan^{-1}(1) \]

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

### Step 5: Adjust for the correct quadrant
Since \(-1 - i\) is in the third quadrant, we need to adjust the angle:
\[ \theta = \frac{\pi}{4} + \pi = \frac{\pi}{4} + \frac{4\pi}{4} = \frac{5\pi}{4} \]

### Step 6: Write the principal argument
Thus, the principal argument of the complex number \(-1 - i\) is:
\[ \text{arg}(z) = \frac{5\pi}{4} \]

### Final Answer
The principal argument of \(-1 - i\) is \(\frac{5\pi}{4}\).

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