The argument of the complex number `(1 +i)^(4)` is

Options

A. `135^(@)`
B. `180^(@)`
C. `90^(@)`
D. `45^(@)`

Answer

To find the argument of the complex number \((1 + i)^4\), we can follow these steps:

### Step 1: Express the complex number in polar form
The complex number \(1 + i\) can be expressed in polar form. We first find the modulus \(r\) and the argument \(\theta\).

The modulus \(r\) is given by:
\[ r = |1 + i| = \sqrt{1^2 + 1^2} = \sqrt{2} \]

The argument \(\theta\) is given by:
\[ \theta = \tan^{-1}\left(\frac{\text{Imaginary part}}{\text{Real part}}\right) = \tan^{-1}\left(\frac{1}{1}\right) = \tan^{-1}(1) = \frac{\pi}{4} \]

### Step 2: Write the complex number in polar form
Using the modulus and argument, we can write:
\[ 1 + i = \sqrt{2}\left(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}\right) \]

### Step 3: Raise the complex number to the power of 4
Using De Moivre's theorem, we can raise the complex number to the power of 4:
\[ (1 + i)^4 = \left(\sqrt{2}\right)^4 \left(\cos\left(4 \cdot \frac{\pi}{4}\right) + i\sin\left(4 \cdot \frac{\pi}{4}\right)\right) \]
Calculating \(\left(\sqrt{2}\right)^4\):
\[ \left(\sqrt{2}\right)^4 = 2^2 = 4 \]
Calculating the angles:
\[ 4 \cdot \frac{\pi}{4} = \pi \]
Thus, we have:
\[ (1 + i)^4 = 4\left(\cos \pi + i\sin \pi\right) \]

### Step 4: Simplify the expression
Using the values of cosine and sine:
\[ \cos \pi = -1 \quad \text{and} \quad \sin \pi = 0 \]
So we get:
\[ (1 + i)^4 = 4(-1 + 0i) = -4 \]

### Step 5: Find the argument of the resulting complex number
The complex number \(-4\) can be expressed as:
\[ -4 + 0i \]
The argument of a complex number in the form \(x + yi\) is given by:
\[ \theta = \tan^{-1}\left(\frac{y}{x}\right) \]
In our case:
\[ \theta = \tan^{-1}\left(\frac{0}{-4}\right) \]
Since the imaginary part is \(0\) and the real part is negative, the argument is:
\[ \theta = \pi \quad \text{(or 180 degrees)} \]

### Final Answer
Thus, the argument of the complex number \((1 + i)^4\) is:
\[ \boxed{\pi} \]