GeoGebra

Complex Cosine

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Complex Cosine

These files show the images of various lines under the complex mapping f(z) = cos(z). All of them are, essentially, applications of the complex powers of e. The constructions are based on cos(z)=\frac{e^{iz}+e^{-iz}}{2}.

Image of a Horizontal Line

The Ggb.gifimage of a horizontal line is an ellipse whose foci are (1,0) and (-1,0). As z moves along this line, cos(z) makes one complete revolution about the ellipse for every increment of that z moves horizontally.

Image of a Vertical Line

The Ggb.gifimage of a vertical line is one branch of a hyperbola whose foci are (1,0) and (-1,0). As z moves along the vertical line, the hyperbola oscilates back and forth between these two foci, making one complete oscilation for every increment of that z moves along the vertical line.

Image of an Arbitrary Line

The Ggb.gifimage of an arbitrary line is just a very pretty curve!

Image of the x Axis

When z is restricted to the x-axis, the cos(z) oscilates between (-1,0) and (1,0) along the x-axis. This file shows the Ggb.gifrelationship between the x-coordinates of z and of cos(z)and how it produces the familiar cosine graph.