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

Options

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

Explanation

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 terms in the numerator:
\[ (1+i)(2+i) = 1 \cdot 2 + 1 \cdot i + i \cdot 2 + i \cdot i = 2 + i + 2i + i^2 \]
Since \(i^2 = -1\), we can simplify:
\[ = 2 + 3i - 1 = 1 + 3i \]

### Step 2: Write the Complex Number
Now, we can rewrite the complex number as:
\[ \frac{1 + 3i}{3 - i} \]

### Step 3: Rationalize the Denominator
To simplify this expression, we will multiply the numerator and the denominator by the conjugate of the denominator, \(3 + i\):
\[ \frac{(1 + 3i)(3 + i)}{(3 - i)(3 + i)} \]

### Step 4: Simplify the Denominator
Calculating the denominator:
\[ (3 - i)(3 + i) = 3^2 - i^2 = 9 - (-1) = 9 + 1 = 10 \]

### Step 5: Simplify the Numerator
Now, calculating the numerator:
\[ (1 + 3i)(3 + i) = 1 \cdot 3 + 1 \cdot i + 3i \cdot 3 + 3i \cdot i = 3 + i + 9i + 3i^2 \]
Again, substituting \(i^2 = -1\):
\[ = 3 + 10i - 3 = 10i \]

### Step 6: Combine the Results
Putting it all together, we have:
\[ \frac{10i}{10} = i \]

### Step 7: Find the Argument
The complex number \(i\) can be expressed in rectangular form as \(0 + 1i\). The argument of a complex number \(z = a + bi\) is given by:
\[ \text{arg}(z) = \tan^{-1}\left(\frac{b}{a}\right) \]
In this case, \(a = 0\) and \(b = 1\):
\[ \text{arg}(i) = \tan^{-1}\left(\frac{1}{0}\right) \]
This is undefined, but we know that the argument of \(i\) corresponds to \(\frac{\pi}{2}\) radians.

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

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