The complex number `z=(i-1)/(cos((pi)/(3))+i sin((pi)/(3)))` is equal to :

Options

A. `sqrt(2)i(cos(5 pi)/(12)-i sin(5 pi)/(12))`
B. `cos(pi)/(12)-i sin(pi)/(12)`
C. `sqrt(2)(cos(pi)/(12)+i sin(pi)/(12))`
D. `sqrt(2)(cos(5 pi)/(12)+i sin(5 pi)/(12))`

Answer

To solve the problem, we need to simplify the complex number \( z = \frac{i - 1}{\cos\left(\frac{\pi}{3}\right) + i \sin\left(\frac{\pi}{3}\right)} \).

### Step 1: Evaluate the denominator
First, we evaluate the trigonometric functions in the denominator:
\[ \cos\left(\frac{\pi}{3}\right) = \frac{1}{2}, \quad \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \]
Thus, the denominator becomes:
\[ \cos\left(\frac{\pi}{3}\right) + i \sin\left(\frac{\pi}{3}\right) = \frac{1}{2} + i \frac{\sqrt{3}}{2} \]

### Step 2: Rewrite the complex number
Now we can rewrite \( z \):
\[ z = \frac{i - 1}{\frac{1}{2} + i \frac{\sqrt{3}}{2}} \]

### Step 3: Rationalize the denominator
To simplify \( z \), we multiply the numerator and the denominator by the conjugate of the denominator:
\[ z = \frac{(i - 1) \left(\frac{1}{2} - i \frac{\sqrt{3}}{2}\right)}{\left(\frac{1}{2} + i \frac{\sqrt{3}}{2}\right)\left(\frac{1}{2} - i \frac{\sqrt{3}}{2}\right)} \]

### Step 4: Calculate the denominator
Calculating the denominator:
\[ \left(\frac{1}{2}\right)^2 + \left(\frac{\sqrt{3}}{2}\right)^2 = \frac{1}{4} + \frac{3}{4} = 1 \]
So the denominator simplifies to 1.

### Step 5: Expand the numerator
Now we expand the numerator:
\[ (i - 1) \left(\frac{1}{2} - i \frac{\sqrt{3}}{2}\right) = \frac{1}{2}i - \frac{1}{2} - i \frac{\sqrt{3}}{2} + \frac{\sqrt{3}}{2} \]
Combining like terms:
\[ = \left(\frac{1}{2} + \frac{\sqrt{3}}{2}\right) + \left(\frac{1}{2} - \frac{\sqrt{3}}{2}\right)i \]

### Step 6: Write in standard form
Thus, we have:
\[ z = \left(\frac{1}{2} + \frac{\sqrt{3}}{2}\right) + \left(\frac{1}{2} - \frac{\sqrt{3}}{2}\right)i \]

### Step 7: Find the modulus and argument
Now we can express \( z \) in polar form. The modulus \( r \) is given by:
\[ r = \sqrt{\left(\frac{1}{2} + \frac{\sqrt{3}}{2}\right)^2 + \left(\frac{1}{2} - \frac{\sqrt{3}}{2}\right)^2} \]
The argument \( \theta \) can be found using:
\[ \tan(\theta) = \frac{\text{Imaginary part}}{\text{Real part}} = \frac{\frac{1}{2} - \frac{\sqrt{3}}{2}}{\frac{1}{2} + \frac{\sqrt{3}}{2}} \]

### Step 8: Simplify to find \( \theta \)
Calculating this gives:
\[ \tan(\theta) = \frac{\frac{1 - \sqrt{3}}{2}}{\frac{1 + \sqrt{3}}{2}} = \frac{1 - \sqrt{3}}{1 + \sqrt{3}} \]
This simplifies to \( \tan\left(\frac{5\pi}{12}\right) \).

### Final Answer
Thus, the value of \( z \) is:
\[ \theta = \frac{5\pi}{12} \]