Complex Cosine Function/Examples/4 cos z = 3+i

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Example of Complex Cosine Function

Let:

$4 \cos z = 3 + i$

Then:

$z = \dfrac {\paren {8 n + 1} \pi} 4 - \dfrac {i \ln 2} 2$ for $n \in \Z$

or:

$z = \dfrac {\paren {8 m - 1} i \pi} 4 + \dfrac {i \ln 2} 2$ for $m \in \Z$


Proof

\(\displaystyle 4 \cos z\) \(=\) \(\displaystyle 3 + i\) $\quad$ $\quad$
\(\displaystyle \leadsto \ \ \) \(\displaystyle 4 \paren {\dfrac {e^{i z} + e^{-i z} } 2}\) \(=\) \(\displaystyle 3 + i\) $\quad$ Cosine Exponential Formulation $\quad$
\(\displaystyle \leadsto \ \ \) \(\displaystyle 2 e^{2 i z} - \paren {3 + i} e^{i z} + 2\) \(=\) \(\displaystyle 0\) $\quad$ mulitplying by $e^{i z}$ and rearranging $\quad$
\(\displaystyle \leadsto \ \ \) \(\displaystyle e^{i z}\) \(=\) \(\displaystyle \dfrac {3 + i \pm \sqrt {\paren {3 + i}^2 - 4 \times 2 \times 2} } {2 \times 2}\) $\quad$ Quadratic Formula $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {3 + i \pm \sqrt {9 - 1 + 6 i - 16} } 4\) $\quad$ Square of Sum $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {3 + i \pm \sqrt {-8 + 6 i} } 4\) $\quad$ simplification $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {3 + i \pm \paren {1 + 3 i} } 4\) $\quad$ Square Root of $-8 + 6 i$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {4 + 4 i} 4 \text { or } \dfrac {2 - 2 i} 4\) $\quad$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle 1 + i \text { or } \dfrac 1 2 - \dfrac i 2\) $\quad$ $\quad$


Thus we have:

\(\displaystyle e^{i z}\) \(=\) \(\displaystyle 1 + i\) $\quad$ $\quad$
\(\displaystyle \leadsto \ \ \) \(\displaystyle i z\) \(=\) \(\displaystyle \ln \paren {1 + i}\) $\quad$ Definition 2 of Complex Logarithm $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \ln \paren {\sqrt 2 \exp \paren {\dfrac {i \pi} 4} }\) $\quad$ Definition of Exponential Form of Complex Number $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \ln \sqrt 2 + \dfrac {i \pi} 4 + 2 n i \pi\) $\quad$ Definition 1 of Complex Logarithm $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \dfrac {\ln 2} 2 + \dfrac {\paren {8 n + 1} i \pi} 4\) $\quad$ simplification $\quad$
\(\displaystyle \leadsto \ \ \) \(\displaystyle z\) \(=\) \(\displaystyle -\dfrac {i \ln 2} 2 + \dfrac {\paren {8 n + 1} \pi} 4\) $\quad$ multiplying through by $-i = \dfrac 1 i$ $\quad$


and:

\(\displaystyle e^{i z}\) \(=\) \(\displaystyle \dfrac 1 2 - \dfrac i 2\) $\quad$ $\quad$
\(\displaystyle \leadsto \ \ \) \(\displaystyle i z\) \(=\) \(\displaystyle \ln \paren {\dfrac 1 2 - \dfrac i 2}\) $\quad$ Definition 2 of Complex Logarithm $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \ln \paren {\dfrac 2 {\sqrt 2} \exp \paren {-\dfrac \pi 4} }\) $\quad$ Definition of Exponential Form of Complex Number $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \ln \dfrac 2 {\sqrt 2} + \dfrac {-i \pi} 4 + 2 m i \pi\) $\quad$ Definition 1 of Complex Logarithm $\quad$
\(\displaystyle \) \(=\) \(\displaystyle -\dfrac {\ln 2} 2 + \dfrac {\paren {8 m - 1} i \pi} 4\) $\quad$ simplification $\quad$
\(\displaystyle \leadsto \ \ \) \(\displaystyle z\) \(=\) \(\displaystyle \dfrac {i \ln 2} 2 + \dfrac {\paren {8 m - 1} \pi} 4\) $\quad$ multiplying through by $-i = \dfrac 1 i$ $\quad$

$\blacksquare$


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