Ellipse and Hyperbola Loci of Complex Cosine Function

This GeoGebra example demonstrates how the complex cosine function w = cos(z) maps vertical lines in the complex plane into families of ellipses and hyperbolas. Key mathematical concepts include the complex cosine formula cos(z) = (e^iz + e^-iz)/2, standard equations of ellipses and hyperbolas, exponential representation of complex numbers, and locus generation. The construction uses Line commands to create vertical lines as the domain, Point commands to select moving points z₁ on these lines, computes w = cos(z), and utilizes the Locus command to generate the trace of w, showing that vertical lines are transformed into ellipses or hyperbolas under the cosine mapping. Suitable for high school to university students learning introductory complex analysis, parameterization of conic sections in analytic geometry, and mastering GeoGebra locus command techniques.