The complex cosine function

The applet below demonstrates the cosine of complex numbers. Move the sliders on the left to move the point \(z\) around, and watch the effect on \(\cos(z)\). Note what happens when you move just the \(x\) slider or \(y\) slider. You can also drag the \(z\) point around. The point \(\cos(z)\) always lies at the intersection of an ellipse and a hyperbola that both have foci at +1 and -1.

By setting one slider appropriately and moving the other, can you "see" the usual real-valued cosine and hyperbolic cosine? Can you "find" hyperbolic sine and the usual sine functions?

The usual cosine function has a root when \(x\) is \(2\pi n + \pi/2\). Does the full complex cosine function have roots for any other values?

You know that with real numbers, \(\cos(x) = x\) is only true in one place (when \(x\) is about 0.739). Are there other complex numbers where \(\cos(z) = z\) is true?

Dan Drake, Created with GeoGebra

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