The modulus and amplitude of `(1+2i)/(1-(1-i)^(2))` are

Options

A. `sqrt(2)` and `(pi)/(6)`
B. `1` and `(pi)/(4)`
C. `1` and `0`
D. `1` and `(pi)/(3)`

Explanation

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

### Step 1: Simplify the Denominator
First, we need to simplify the denominator \(1 - (1 - i)^2\).

Calculating \((1 - i)^2\):
\[ (1 - i)^2 = 1^2 - 2 \cdot 1 \cdot i + i^2 = 1 - 2i + (-1) = 0 - 2i = -2i \]

Now substitute this back into the denominator:
\[ 1 - (1 - i)^2 = 1 - (-2i) = 1 + 2i \]

### Step 2: Rewrite the Expression
Now we can rewrite the original expression:
\[ \frac{1 + 2i}{1 + 2i} \]

### Step 3: Simplify the Expression
Since the numerator and denominator are the same, we can simplify:
\[ \frac{1 + 2i}{1 + 2i} = 1 \]

### Step 4: Find the Modulus
The modulus of a complex number \(z = a + bi\) is given by \(|z| = \sqrt{a^2 + b^2}\). In this case, since \(z = 1 + 0i\):
\[ |z| = \sqrt{1^2 + 0^2} = \sqrt{1} = 1 \]

### Step 5: Find the Amplitude
The amplitude (or argument) \(\theta\) of a complex number is given by \(\tan \theta = \frac{y}{x}\), where \(y\) is the imaginary part and \(x\) is the real part. Here, \(y = 0\) and \(x = 1\):
\[ \tan \theta = \frac{0}{1} = 0 \]

Thus, the angle \(\theta\) is:
\[ \theta = 0 \]

### Final Result
The modulus of the complex number is \(1\) and the amplitude is \(0\).

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