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#### Petrus

##### Well-known member

- Feb 21, 2013

- 739

calculate \(\displaystyle \left(\frac{1}{2}+i\frac{\sqrt{3}}{2} \right)^{100}\) in the form \(\displaystyle a+ib\)

progress:

I start to calculate argument and get it to \(\displaystyle r=1\) (argument)

then \(\displaystyle \cos\theta=\frac{1}{2} \ sin\theta=\frac{\sqrt{3}}{2}\) we se it's in first quadrant( where x and y is positive)

\(\displaystyle 1*e^{i\frac{100\pi}{3}}\)

notice that we can always take away 2pi so we can simplify that to

\(\displaystyle 1*e^{i\frac{\pi}{3}}\)

\(\displaystyle 1*e^{i\frac{\pi}{3}}=\cos(\frac{\pi}{3}) + i \sin (\frac{\pi}{3}) = \frac{1}{2}+i\frac{\sqrt{3}}{2}\)

but the facit says \(\displaystyle \frac{-1}{2}-i\frac{\sqrt{3}}{2}\)

Regards,