If z is a point on the circle `|z-1|=1,` then arg z =

Options

A. `arg|z-1|`
B. `(1)/(2)arg(z-1)`
C. `arg(z^(2)-z)`
D. `(1)/(3)arg(z^(2)-z)`

Answer

To solve the problem, we need to find the argument of the complex number \( z \) given that \( |z - 1| = 1 \). This equation describes a circle in the complex plane with center at \( 1 \) and radius \( 1 \).

### Step-by-step Solution:

1. **Understanding the Circle**:
The equation \( |z - 1| = 1 \) indicates that \( z \) lies on a circle centered at the point \( 1 \) on the real axis (which corresponds to the complex number \( 1 + 0i \)) with a radius of \( 1 \).

2. **Expressing \( z \)**:
We can express \( z \) in terms of an angle \( \theta \):
\[ z - 1 = e^{i\theta} \quad \text{(where \( \theta \) is the angle in radians)} \]
Thus, we can write:
\[ z = 1 + e^{i\theta} = 1 + \cos(\theta) + i\sin(\theta) \]

3. **Finding the Argument of \( z \)**:
The argument of \( z \) can be found using the formula for the argument of a complex number:
\[ \text{arg}(z) = \tan^{-1}\left(\frac{\text{Im}(z)}{\text{Re}(z)}\right) \]
Here, the real part \( \text{Re}(z) = 1 + \cos(\theta) \) and the imaginary part \( \text{Im}(z) = \sin(\theta) \).

4. **Calculating the Argument**:
Therefore, we have:
\[ \text{arg}(z) = \tan^{-1}\left(\frac{\sin(\theta)}{1 + \cos(\theta)}\right) \]
Using the identity \( 1 + \cos(\theta) = 2\cos^2(\frac{\theta}{2}) \) and \( \sin(\theta) = 2\sin(\frac{\theta}{2}\cos(\frac{\theta}{2}) \), we can rewrite the argument:
\[ \text{arg}(z) = \tan^{-1}\left(\frac{2\sin(\frac{\theta}{2})\cos(\frac{\theta}{2})}{2\cos^2(\frac{\theta}{2})}\right) = \tan^{-1}\left(\tan\left(\frac{\theta}{2}\right)\right) \]
Thus, we find:
\[ \text{arg}(z) = \frac{\theta}{2} \]

5. **Relating to \( \text{arg}(z - 1) \)**:
Since \( z - 1 = e^{i\theta} \), we have:
\[ \text{arg}(z - 1) = \theta \]
Therefore, we can relate the arguments:
\[ \text{arg}(z) = \frac{1}{2} \text{arg}(z - 1) \]

### Final Result:
Thus, we conclude that:
\[ \text{arg}(z) = \frac{1}{2} \text{arg}(z - 1) \]