Principal argument of `z = (i-1)/(i(1-"cos"(2pi)/(7))+"sin"(2pi)/(7))` is

Options

A. `(pi)/(28)`
B. `(3pi)/(28)`
C. `(17 pi)/(28)`
D. `(19 pi)/(28)`

Answer

To find the principal argument of the complex number \( z = \frac{i - 1}{i(1 - \cos(\frac{2\pi}{7})) + \sin(\frac{2\pi}{7})} \), we will follow these steps:

### Step 1: Simplify the Denominator
First, we need to simplify the denominator:
\[ i(1 - \cos(\frac{2\pi}{7})) + \sin(\frac{2\pi}{7}) \]
This can be rewritten as:
\[ i(1 - \cos(\frac{2\pi}{7})) + \sin(\frac{2\pi}{7}) = \sin(\frac{2\pi}{7}) + i(1 - \cos(\frac{2\pi}{7})) \]

### Step 2: Rewrite \( z \)
Now, we can rewrite \( z \):
\[ z = \frac{i - 1}{\sin(\frac{2\pi}{7}) + i(1 - \cos(\frac{2\pi}{7}))} \]

### Step 3: Multiply by the Conjugate
To simplify \( z \), we multiply the numerator and denominator by the conjugate of the denominator:
\[ z = \frac{(i - 1)(\sin(\frac{2\pi}{7}) - i(1 - \cos(\frac{2\pi}{7})))}{(\sin(\frac{2\pi}{7}) + i(1 - \cos(\frac{2\pi}{7})))(\sin(\frac{2\pi}{7}) - i(1 - \cos(\frac{2\pi}{7})))} \]

### Step 4: Calculate the Denominator
The denominator simplifies to:
\[ \sin^2(\frac{2\pi}{7}) + (1 - \cos(\frac{2\pi}{7}))^2 \]
Using the identity \( 1 - \cos^2(x) = \sin^2(x) \), we can express it as:
\[ \sin^2(\frac{2\pi}{7}) + (1 - \cos(\frac{2\pi}{7}))^2 = \sin^2(\frac{2\pi}{7}) + (1 - 2\cos(\frac{2\pi}{7}) + \cos^2(\frac{2\pi}{7})) = 1 - 2\cos(\frac{2\pi}{7}) + 1 = 2(1 - \cos(\frac{2\pi}{7})) \]

### Step 5: Calculate the Numerator
The numerator simplifies to:
\[ (i - 1)(\sin(\frac{2\pi}{7}) - i(1 - \cos(\frac{2\pi}{7})) = i\sin(\frac{2\pi}{7}) - i^2(1 - \cos(\frac{2\pi}{7})) - \sin(\frac{2\pi}{7}) + i \]
Since \( i^2 = -1 \), we have:
\[ = i\sin(\frac{2\pi}{7}) + (1 - \cos(\frac{2\pi}{7})) - \sin(\frac{2\pi}{7}) = (1 - \cos(\frac{2\pi}{7})) + i\sin(\frac{2\pi}{7}) \]

### Step 6: Combine the Results
Now we can write \( z \):
\[ z = \frac{(1 - \cos(\frac{2\pi}{7})) + i\sin(\frac{2\pi}{7})}{2(1 - \cos(\frac{2\pi}{7}))} \]
This simplifies to:
\[ z = \frac{1 - \cos(\frac{2\pi}{7})}{2(1 - \cos(\frac{2\pi}{7}))} + i\frac{\sin(\frac{2\pi}{7})}{2(1 - \cos(\frac{2\pi}{7}))} \]
\[ = \frac{1}{2} + i\frac{\sin(\frac{2\pi}{7})}{2(1 - \cos(\frac{2\pi}{7}))} \]

### Step 7: Find the Argument
The argument of \( z \) is given by:
\[ \text{arg}(z) = \tan^{-1}\left(\frac{\text{Imaginary part}}{\text{Real part}}\right) = \tan^{-1}\left(\frac{\frac{\sin(\frac{2\pi}{7})}{2(1 - \cos(\frac{2\pi}{7})}}{\frac{1}{2}}\right) = \tan^{-1}\left(\frac{\sin(\frac{2\pi}{7})}{1 - \cos(\frac{2\pi}{7})}\right) \]

### Step 8: Use the Tangent Identity
Using the identity \( \tan(\theta) = \frac{\sin(\theta)}{1 - \cos(\theta)} \), we have:
\[ \text{arg}(z) = \tan^{-1}(\tan(\frac{2\pi}{7})) = \frac{2\pi}{7} \]

### Step 9: Determine the Principal Argument
Since the principal argument is typically in the range \( (-\pi, \pi] \), we find:
\[ \text{arg}(z) = \frac{2\pi}{7} \]

### Final Answer
Thus, the principal argument of \( z \) is:
\[ \frac{2\pi}{7} \]