Hi folks,

I am asked to use the definition $\cos z = \dfrac{e^{iz} + e^{-iz}}{2}$ to find 2 imaginary numbers having a cosine of 4.

so, $\cos z = 4$ therefore

$4 = \dfrac{e^{iz} + e^{-iz}}{2} $

$8e^{iz} = e^{i2z} + 1$

let $w = e^{iz}$

$w^2 - 8w + 1 = 0$

$w = 4 \pm \sqrt{15}$

$e^{iz} = 4 \pm \sqrt{15}$

$iz \ln e = \ln (4 \pm \sqrt{15}) $

multiply both sides by i

$i^2 z = i \ln(4 \pm \sqrt 15)$

$ -z = i \ln(4 \pm \sqrt 15)$

$z = i \ln \dfrac{1}{(4 \pm \sqrt 15)}$

the actual answer is

$z = i \ln (4 \pm \sqrt{15}) $

can anyone see what I am doing wrong?