Author Topic: TT1 Problem 1 (noon)  (Read 670 times)

Victor Ivrii

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TT1 Problem 1 (noon)
« on: October 19, 2018, 04:03:25 AM »
Find all the complex roots of the equation $\cos (z) = 3$.

ZhenDi Pan

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Re: TT1 Problem 1 (noon)
« Reply #1 on: October 19, 2018, 05:31:45 AM »
My solution:
\begin{align*}
&\cos(z)=\frac{1}{2}(e^{iz}+e^{-iz})=3 \\
&(e^{iz}+e^{-iz}) = 6 \\
&e^{2iz}+1 =6e^{iz} \\
&e^{2iz}+1-6e^{iz} =0 \\
&e^{iz} = \frac{6\pm \sqrt{36-4}}{2}=3\pm 2\sqrt{2} \\
&iz=\log(3\pm 2\sqrt{2}) \\
&iz=\log(3\pm 2\sqrt{2})=\ln(3\pm 2\sqrt{2})+i2k\pi \qquad  k\in \mathbb{Z} \\
&z = -i\ln(3\pm 2\sqrt{2})+2k\pi \qquad k \in \mathbb{Z}
\end{align*}

Please reformat: \cos, \sin , \ln, \log etc
« Last Edit: October 19, 2018, 06:16:33 AM by ZhenDi Pan »