Complex Arithmetic/Examples/((root 3 - i)(root 3 + i)^-1)^4 ((1 + i)(1 - i)^-1)^5

From ProofWiki
Jump to navigation Jump to search

Example of Complex Arithmetic

$\paren {\dfrac {\sqrt 3 - i} {\sqrt 3 + i} }^4 \paren {\dfrac {1 + i} {1 - i} }^5 = -\dfrac {\sqrt 3} 2 - \dfrac 1 2 i$


Proof

\(\displaystyle \paren {\dfrac {\sqrt 3 - i} {\sqrt 3 + i} }^4 \paren {\dfrac {1 + i} {1 - i} }^5\) \(=\) \(\displaystyle \paren {\dfrac {\paren {\sqrt 3 - i}^2} {\paren {\sqrt 3 + i} \paren {\sqrt 3 - i} } }^4 \paren {\dfrac {\paren {1 + i}^2} {\paren {1 - i} \paren {1 + i} } }^5\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {\dfrac {3 + i^2 - 2 \sqrt 3 i} {\sqrt 3^2 + 1^2} }^4 \paren {\dfrac {1 + i^2 + 2 i} {1^2 + 1^2} }^5\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {\dfrac {2 - 2 \sqrt 3 i} 4}^4 \paren {\dfrac {2 i} 2}^5\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {\dfrac 1 2 - \dfrac {\sqrt 3} 2 i}^4 i^5\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {\cis \dfrac {5 \pi} 3}^4 i\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {\cis \dfrac {20 \pi} 3} i\) De Moivre's Formula
\(\displaystyle \) \(=\) \(\displaystyle \paren {\cis \dfrac {18 \pi} 3} \paren {\cis \dfrac {2 \pi} 3} i\)
\(\displaystyle \) \(=\) \(\displaystyle \paren {\cis \dfrac {2 \pi} 3} \paren {\cis \dfrac \pi 2}\)
\(\displaystyle \) \(=\) \(\displaystyle \cis \dfrac {7 \pi} 6\)
\(\displaystyle \) \(=\) \(\displaystyle \cos \dfrac {7 \pi} 6 + i \sin \dfrac {7 \pi} 6\)
\(\displaystyle \) \(=\) \(\displaystyle -\dfrac {\sqrt 3} 2 - \dfrac 1 2 i\) Cosine of $210 \degrees$ and Sine of $210 \degrees$

$\blacksquare$


Sources