What is the argument of the complex number `((1+i)(2+i))/(3-i)`, where `i = sqrt(-1)` ?

Options

A. 0
B. `(pi)/(4)`
C. `- ( pi )/( 4)`
D. `(pi)/(2)`

Answer

To find the argument of the complex number \(\frac{(1+i)(2+i)}{3-i}\), we will follow these steps:

### Step 1: Multiply the Numerator
First, we need to multiply the two complex numbers in the numerator:

\[ (1+i)(2+i) = 1 \cdot 2 + 1 \cdot i + i \cdot 2 + i \cdot i \]

Calculating this gives:

\[ = 2 + i + 2i + i^2 \]
Since \(i^2 = -1\), we have:

\[ = 2 + 3i - 1 = 1 + 3i \]

### Step 2: Rationalize the Denominator
Next, we need to rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator:

\[ \frac{(1 + 3i)(3 + i)}{(3 - i)(3 + i)} \]

Calculating the denominator:

\[ (3 - i)(3 + i) = 3^2 + 1^2 = 9 + 1 = 10 \]

Now, calculating the numerator:

\[ (1 + 3i)(3 + i) = 1 \cdot 3 + 1 \cdot i + 3i \cdot 3 + 3i \cdot i \]
This expands to:

\[ = 3 + i + 9i + 3i^2 \]
Again, since \(i^2 = -1\):

\[ = 3 + 10i - 3 = 10i \]

### Step 3: Combine the Results
Now we can combine the results:

\[ \frac{10i}{10} = i \]

### Step 4: Find the Argument
The complex number \(i\) can be expressed as \(0 + 1i\). The argument of a complex number in the form \(x + yi\) is given by:

\[ \text{arg}(z) = \tan^{-1}\left(\frac{y}{x}\right) \]

In our case, \(x = 0\) and \(y = 1\):

\[ \text{arg}(i) = \tan^{-1}\left(\frac{1}{0}\right) \]

Since \(\frac{1}{0}\) is undefined, the argument corresponds to \(\frac{\pi}{2}\) (90 degrees).

### Final Answer
Thus, the argument of the complex number \(\frac{(1+i)(2+i)}{3-i}\) is:

\[ \frac{\pi}{2} \]