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

## 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

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

Thus we have:

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

and:

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

$\blacksquare$