What is `((sin ""(pi)/( 6) +i (1-cos""(pi)/(6)))/(sin""(pi)/(6)-i(1-cos""(pi)/(6))))^(3)` is equal to ?

Options

A. 1
B. `-1`
C. i
D. `-i`

Answer

To solve the problem, we need to evaluate the expression:

\[ \left( \frac{\sin\left(\frac{\pi}{6}\right) + i(1 - \cos\left(\frac{\pi}{6}\right))}{\sin\left(\frac{\pi}{6}\right) - i(1 - \cos\left(\frac{\pi}{6}\right))} \right)^3 \]

### Step 1: Calculate \(\sin\left(\frac{\pi}{6}\right)\) and \(\cos\left(\frac{\pi}{6}\right)\)

Using known values:
\[ \sin\left(\frac{\pi}{6}\right) = \frac{1}{2} \]
\[ \cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \]

### Step 2: Substitute these values into the expression

Substituting the values into the expression gives:
\[ \frac{\frac{1}{2} + i\left(1 - \frac{\sqrt{3}}{2}\right)}{\frac{1}{2} - i\left(1 - \frac{\sqrt{3}}{2}\right)} \]

### Step 3: Simplify the numerator and denominator

The numerator simplifies to:
\[ \frac{1}{2} + i\left(1 - \frac{\sqrt{3}}{2}\right) = \frac{1}{2} + i\left(\frac{2 - \sqrt{3}}{2}\right) = \frac{1 + i(2 - \sqrt{3})}{2} \]

The denominator simplifies to:
\[ \frac{1}{2} - i\left(1 - \frac{\sqrt{3}}{2}\right) = \frac{1}{2} - i\left(\frac{2 - \sqrt{3}}{2}\right) = \frac{1 - i(2 - \sqrt{3})}{2} \]

### Step 4: Rewrite the expression

Now we can rewrite the expression:
\[ \frac{\frac{1 + i(2 - \sqrt{3})}{2}}{\frac{1 - i(2 - \sqrt{3})}{2}} = \frac{1 + i(2 - \sqrt{3})}{1 - i(2 - \sqrt{3})} \]

### Step 5: Multiply numerator and denominator by the conjugate of the denominator

To simplify further, multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{(1 + i(2 - \sqrt{3}))(1 + i(2 - \sqrt{3}))}{(1 - i(2 - \sqrt{3}))(1 + i(2 - \sqrt{3}))} \]

### Step 6: Simplify the denominator

The denominator simplifies to:
\[ 1^2 + (2 - \sqrt{3})^2 = 1 + (4 - 4\sqrt{3} + 3) = 8 - 4\sqrt{3} \]

### Step 7: Expand the numerator

The numerator expands to:
\[ (1 + i(2 - \sqrt{3}))^2 = 1 + 2i(2 - \sqrt{3}) - (2 - \sqrt{3})^2 = 1 + 2i(2 - \sqrt{3}) - (4 - 4\sqrt{3} + 3) = -6 + 4\sqrt{3} + 2i(2 - \sqrt{3}) \]

### Step 8: Combine and simplify

Now we have:
\[ \frac{-6 + 4\sqrt{3} + 2i(2 - \sqrt{3})}{8 - 4\sqrt{3}} \]

### Step 9: Evaluate the cube of the expression

To find the cube of this expression, we can convert it to polar form and apply De Moivre's theorem. However, we can also use the fact that the expression simplifies to \(e^{i\theta}\) for some angle \(\theta\).

### Step 10: Final evaluation

After evaluating the angle, we find that:
\[ \left( e^{i\frac{\pi}{2}} \right)^3 = e^{i\frac{3\pi}{2}} = -i \]

Thus, the final answer is:
\[ \boxed{-i} \]