If ` Z = (1+7i)/((2-i)^(2))`then the polar form of z is

Options

A. `sqrt2(cos"(3pi)/4-isin"" (3pi)/4)`
B. `sqrt2(cos"(3pi)/4+isin"" (3pi)/4)`
C. `sqrt2(cos"(7pi)/4+isin"" (7pi)/4)`
D. `sqrt2(cos"(7pi)/4-isin"" (7pi)/4)`

Answer

To find the polar form of the complex number \( Z = \frac{1 + 7i}{(2 - i)^2} \), we will follow these steps:

### Step 1: Calculate \( (2 - i)^2 \)
First, we need to square the denominator:
\[ (2 - i)^2 = 2^2 - 2 \cdot 2 \cdot i + i^2 = 4 - 4i + (-1) = 3 - 4i \]

### Step 2: Rewrite \( Z \)
Now, we can rewrite \( Z \) using the result from Step 1:
\[ Z = \frac{1 + 7i}{3 - 4i} \]

### Step 3: Rationalize the denominator
To simplify \( Z \), we multiply the numerator and the denominator by the conjugate of the denominator:
\[ Z = \frac{(1 + 7i)(3 + 4i)}{(3 - 4i)(3 + 4i)} \]

### Step 4: Calculate the denominator
Now, calculate the denominator:
\[ (3 - 4i)(3 + 4i) = 3^2 - (4i)^2 = 9 - 16(-1) = 9 + 16 = 25 \]

### Step 5: Calculate the numerator
Next, calculate the numerator:
\[ (1 + 7i)(3 + 4i) = 1 \cdot 3 + 1 \cdot 4i + 7i \cdot 3 + 7i \cdot 4i = 3 + 4i + 21i + 28i^2 \]
Since \( i^2 = -1 \):
\[ = 3 + 25i - 28 = -25 + 25i \]

### Step 6: Combine results
Now, combine the results:
\[ Z = \frac{-25 + 25i}{25} = -1 + i \]

### Step 7: Convert to polar form
To convert \( Z = -1 + i \) to polar form, we need to find the modulus \( r \) and the argument \( \theta \).

1. **Calculate the modulus \( r \)**:
\[ r = |Z| = \sqrt{(-1)^2 + (1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

2. **Calculate the argument \( \theta \)**:
The argument \( \theta \) can be found using:
\[ \tan(\theta) = \frac{\text{Imaginary part}}{\text{Real part}} = \frac{1}{-1} = -1 \]
This corresponds to an angle in the second quadrant since the real part is negative and the imaginary part is positive. Therefore:
\[ \theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \]

### Step 8: Write in polar form
Thus, the polar form of \( Z \) is:
\[ Z = r(\cos \theta + i \sin \theta) = \sqrt{2} \left( \cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4} \right) \]

### Final Answer
The polar form of \( Z \) is:
\[ Z = \sqrt{2} \left( \cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4} \right) \]

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