Labels:circle | graphics | sketch | clipart | art OCR: Figure 12-9: Illustration of the Range of the Cosine Function We are used to seeing cos looking exactly like sin, a wiggly ocean wave, only displaced. Indeed the complex mapping of cos is also similar to that of sin, with horizontal and vertical lines mapping to the same ellipses and hyperbolas with foci at +1, although mapping to them in a diferent manner, to be sure. The entire real axis is again mapped to the segment [-1, 1] of the real axis, but each half of the imaginary axis is mapped to the real axis to the right of 1 (as if by cosh considered as a real function). Therefore Loos both map to +00. The origin is mapped to 1. Whereas sin is an odd function, coe is an even function; as a result two points in each annulus, one the negative of the other, are mapped to the same shaded point in this graph; the shading shown here is taken from points in the original upper half-plane.
