What is the principle argument of `( -1-i)` where `i = sqrt( - 1)`.

Options

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

Answer

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

### Step 1: Identify the real and imaginary parts
The complex number can be expressed in the form \(z = x + iy\), where:
- \(x\) is the real part
- \(y\) is the imaginary part

For \(-1 - i\):
- Real part \(x = -1\)
- Imaginary part \(y = -1\)

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

### Step 3: Calculate the tangent of the angle
The argument \(\theta\) can be found using the formula:
\[ \tan(\theta) = \frac{y}{x} \]
Substituting the values:
\[ \tan(\theta) = \frac{-1}{-1} = 1 \]

### Step 4: Find the reference angle
The angle whose tangent is 1 is:
\[ \theta = 45^\circ \text{ or } \frac{\pi}{4} \text{ radians} \]

### Step 5: Adjust for the correct quadrant
Since the complex number is in the third quadrant, we need to add \(\pi\) to the reference angle:
\[ \theta = \pi + \frac{\pi}{4} = \frac{4\pi}{4} + \frac{\pi}{4} = \frac{5\pi}{4} \]

### Final Answer
The principal argument of \(-1 - i\) is:
\[ \theta = \frac{5\pi}{4} \]

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