obliczyc wartosc wyrazenia

[milagros111]

obliczyc wartosc wyrazenia

\(\displaystyle{ \left[ \frac{1-i}{ \sqrt{3}+i }\right] ^{30}}\)

[gajatko]

O ile się nigdzie nie pomyliłem to powinno to wyglądać tak:
\(\displaystyle{ \left[ \frac{1-i}{ \sqrt{3}+i }\right] ^{30}=
ft[ \frac{(1-i)(\sqrt {3}-i)}{4}\right] ^{30}=
ft[{1 \over 2}\cdot {2 \over \sqrt 2}\left({\sqrt2 \over 2}-{\sqrt2 \over 2}i\right)\left({\sqrt3 \over 2}-{1\over 2}i\right)\right]^{30}=\\
=\left[{+1 \over \sqrt 2}\left(\sin (-{\pi\over 4})+i\cos(-{\pi\over 4})\right)
ft(\sin(-{\pi\over 3})+i\cos(-{\pi\over 3})\right)\right]^{30}=\\
=\left({1 \over \sqrt 2}\right)^{30}\left(\sin (-{{30\pi}\over 4})+i\cos (-{{30\pi}\over 4})\right)\left(\sin (-{{30\pi}\over 3})+i\cos (-{{30\pi}\over 3})\right)=\\
=\left({1 \over \sqrt 2}\right)^{30}\left(\sin (-{{3\pi}\over 2})+i\cos (-{{3\pi}\over 2})\right)\left(\sin 0+i\cos 0\right)=\left({1 \over \sqrt 2}\right)^{30}(-1)(1)=-2^{-15}}\)

[milagros111]

dziekuje:)