gin” and a polar angle ( ). This method
also uses trigonometric functions such as
sin and cos (sine and cosine; for more
about such functions, see trigonometry
in this chapter).
Polar coordinates in three-dimension-
al space—also called spherical coordi-
nates—use rand two polar angles ( , ) to
give the direction from the origin to the
point. To compare, a three-dimensional
polar coordinate system overlaps the
Cartesian system in several ways: For
example, is the angle between the line to
the origin and the z-axis of the Cartesian
(x, y, z) system; is the angle (counter-
clockwise when viewed from positive z)
between the projection of that line onto
the (x, y) plane and the x-axis.
What is an Argand diagram?
An Argand diagram is a graphical way of representing a function of a complex variable,
often written as zxiy,in which x, y,and zare coordinates in three-dimensional
space and iis an imaginary number. Its true discoverer is not actually known, but
Swiss mathematician Jean Robert Argand (1768–1822) is given credit for the diagram.
It is thought that this was also independently discovered by Danish mathematician
Casper Wessel (1745–1818), and later by German mathematician and physicist Karl
Friedrich Gauss (1777–1855) in 1832 (but he probably determined it much earlier);
thus, its other name is the Gaussian plane.
What is an asymptotic curve?
On a graph, a line that approaches close to a curve (or even an axis) but never quite
reaches it is an asymptotic curve. In an example similar to one of Zeno’s paradoxes
(see “Foundations of Mathematics”), if a kitten standing a yard from a box walks half
the distance to the box each hour, it will technically never reach the box, because the
distance it travels each hour is never more than half the remaining distance to the
box. If this problem was illustrated as an equation, the answer would never quite reach
its solution. A more mathematical example is the exponential function y 2 x, which
results in a line that approaches but will never quite reach the x-axis.
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GEOMETRY AND TRIGONOMETRY
In the above illustration, x r cos ; y r sin ; r^2
x^2 y^2 ; and
arctan y/x (x0).