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. First, we need to find its modulus and argument.

**Modulus:**
\[ |1 + i| = \sqrt{1^2 + 1^2} = \sqrt{2} \]

**Argument:**
The argument \(\theta\) can be found using:
\[ \theta = \tan^{-1}\left(\frac{\text{Imaginary part}}{\text{Real part}}\right) = \tan^{-1}\left(\frac{1}{1}\right) = \frac{\pi}{4} \]

Thus, we can express \(1 + i\) in polar form as:
\[ 1 + i = \sqrt{2} \left(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}\right) \]

### Step 2: Raise to the power of 4
Now we need to raise this expression 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 angle:
\[ 4 \cdot \frac{\pi}{4} = \pi \]

Thus, we have:
\[ (1 + i)^4 = 4 \left(\cos \pi + i \sin \pi\right) \]

### Step 3: Simplify using trigonometric values
Using the values of \(\cos \pi\) and \(\sin \pi\):
\[ \cos \pi = -1, \quad \sin \pi = 0 \]

So,
\[ (1 + i)^4 = 4(-1 + 0i) = -4 \]

### Step 4: Find the argument of the resulting complex number
The complex number \(-4\) can be expressed as:
\[ -4 + 0i \]

The argument of a complex number \(a + bi\) is given by:
\[ \text{Argument} = \tan^{-1}\left(\frac{b}{a}\right) \]

For \(-4 + 0i\):
\[ \text{Argument} = \tan^{-1}\left(\frac{0}{-4}\right) \]

Since the point \(-4\) lies on the negative real axis, its argument is:
\[ \pi \quad \text{(or 180 degrees)} \]

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