What is \[ \left( \frac{\sin \frac{\pi}{6} + i(1 - \cos \frac{\pi}{6})}{\sin \frac{\pi}{6} - i(1 - \cos \frac{\pi}{6})} \right)^{3} \] where i = \sqrt{-1}, equal to?

Steps to Solve

1. Calculate sine and cosine values To start, we need to calculate the values of $\sin\left(\frac{\pi}{6}\right)$ and $\cos\left(\frac{\pi}{6}\right)$.

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

2. Substitute values into the expression Now substitute these values into the given expression:

Evaluate (1 - \cos\left(\frac{\pi}{6}\right)):

[ 1 - \cos\left(\frac{\pi}{6}\right) = 1 - \frac{\sqrt{3}}{2} = \frac{2 - \sqrt{3}}{2} ]

The expression inside the brackets becomes:

[ \frac{\sin(\frac{\pi}{6}) + i(1 - \cos(\frac{\pi}{6}))}{\sin(\frac{\pi}{6}) - i(1 - \cos(\frac{\pi}{6}))} = \frac{\frac{1}{2} + i\left(\frac{2 - \sqrt{3}}{2}\right)}{\frac{1}{2} - i\left(\frac{2 - \sqrt{3}}{2}\right)} ]

3. Multiply numerator and denominator by 2 To simplify, multiply both the numerator and denominator by 2:

[ \frac{1 + i(2 - \sqrt{3})}{1 - i(2 - \sqrt{3})} ]

4. Use the formula for division of complex numbers Using the formula for division of complex numbers, we 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}))} ]

5. Simplify The denominator simplifies as follows:

[ (1)^2 + (2 - \sqrt{3})^2 = 1 + (4 - 4\sqrt{3} + 3) = 4 - 4\sqrt{3} + 4 = 8 - 4\sqrt{3} ]

The numerator simplifies using the square of a binomial:

[ (1)^2 + (2 - \sqrt{3})^2 = 1 + (2(2 - \sqrt{3})i) = 1 - (2 - \sqrt{3})^2 = 1 + (4 - 4\sqrt{3} + 3)i^2 ]

6. Calculate the cube Now we cube this expression.

Assume the previous result gives us a complex number $z = a + bi$, so $z^3 = (a + bi)^3$ can be expanded using binomial expansion, or the specific values for $z$ can be calculated to determine the real and imaginary parts.

7. Identify the final result The evaluation leads you to the various possible answers, then select the one that matches.