If `Z=cos phi+isin phi(AA phi in ((pi)/(3),pi))`, then the value of `arg(Z^(2)-Z)` is equal to (where, `arg(Z)` represents the argument of the complex number Z lying in the interval `(-pi, pi] and i^(2)=-1`)

Options

A. `(3phi+pi)/(2)`
B. `(3phi)/(2)`
C. `(3)/(2)(phi-pi)`
D. `(3phi-pi)/(2)`

Answer

To find the value of \( \arg(Z^2 - Z) \) where \( Z = \cos \phi + i \sin \phi \) for \( \phi \in \left(\frac{\pi}{3}, \pi\right) \), we will follow these steps:

### Step 1: Calculate \( Z^2 \)
Given:
\[ Z = \cos \phi + i \sin \phi \]
We can calculate \( Z^2 \):
\[ Z^2 = (\cos \phi + i \sin \phi)^2 = \cos^2 \phi + 2i \cos \phi \sin \phi + i^2 \sin^2 \phi \]
Since \( i^2 = -1 \):
\[ Z^2 = \cos^2 \phi - \sin^2 \phi + 2i \cos \phi \sin \phi \]
Using the double angle identities:
\[ Z^2 = \cos(2\phi) + i \sin(2\phi) \]

### Step 2: Calculate \( Z^2 - Z \)
Now, we compute \( Z^2 - Z \):
\[ Z^2 - Z = \left(\cos(2\phi) + i \sin(2\phi)\right) - \left(\cos \phi + i \sin \phi\right) \]
This simplifies to:
\[ Z^2 - Z = \left(\cos(2\phi) - \cos \phi\right) + i \left(\sin(2\phi) - \sin \phi\right) \]

### Step 3: Find the argument of \( Z^2 - Z \)
We need to find \( \arg(Z^2 - Z) \):
\[ \arg(Z^2 - Z) = \tan^{-1}\left(\frac{\sin(2\phi) - \sin \phi}{\cos(2\phi) - \cos \phi}\right) \]

### Step 4: Simplify the expressions
Using the sine and cosine subtraction formulas:
\[ \sin(2\phi) - \sin(\phi) = 2 \cos\left(\frac{3\phi}{2}\right) \sin\left(\frac{\phi}{2}\right) \]
\[ \cos(2\phi) - \cos(\phi) = -2 \sin\left(\frac{3\phi}{2}\right) \sin\left(\frac{\phi}{2}\right) \]
Thus:
\[ \frac{\sin(2\phi) - \sin(\phi)}{\cos(2\phi) - \cos(\phi)} = \frac{2 \cos\left(\frac{3\phi}{2}\right) \sin\left(\frac{\phi}{2}\right)}{-2 \sin\left(\frac{3\phi}{2}\right) \sin\left(\frac{\phi}{2}\right)} \]
This simplifies to:
\[ -\frac{\cos\left(\frac{3\phi}{2}\right)}{\sin\left(\frac{3\phi}{2}\right)} = -\cot\left(\frac{3\phi}{2}\right) \]

### Step 5: Determine the quadrant
Given \( \phi \in \left(\frac{\pi}{3}, \pi\right) \), we find \( \frac{3\phi}{2} \) will lie in the interval \( \left(\frac{\pi}{2}, \frac{3\pi}{2}\right) \). Therefore, \( -\cot\left(\frac{3\phi}{2}\right) \) will yield a positive value in the third quadrant.

### Step 6: Final argument calculation
Thus, we have:
\[ \arg(Z^2 - Z) = \tan^{-1}\left(-\cot\left(\frac{3\phi}{2}\right)\right) = -\frac{3\phi}{2} + \pi \]

### Conclusion
The final value of \( \arg(Z^2 - Z) \) is:
\[ \arg(Z^2 - Z) = \frac{3\phi}{2} - \pi \]