Friday, September 8, 2017

analysis - Real and imaginary part of $ (1-isqrt{3})^6$




i am a bit stuck here.



As the title says i try to find out how to write complex numbers like for example$$ (1-i\sqrt{3})^6$$ in the normal representation$$ z = x + i*y$$
I already found out that the polar representation of complex numbers will come in handy here, but i can't make the conclusion at the moment.



How can i get from here to the polar representation? How do i get the real and imaginary part from the polar representation? If you have a hint, can you please just leave a quick post here, thanks.


Answer



The modulus of $1-i\sqrt{3}$ is $\sqrt{1+3}=2$, so you can write
$$
1-i\sqrt{3}=2\left(\frac{1}{2}-i\frac{\sqrt{3}}{2}\right)

=
2(\cos(-\pi/3)+i\sin(-\pi/3))
$$
Can you compute the sixth power, now?


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