Polar and cylindrical coordinates/Angle naive/Multiplication in C/Introduction/Section

We discuss several important applications of trigonometric functions like polar coordinates, understanding angles and the trigonometric functions in a naive way.

Example

An angle ${}\alpha$ and a positive real number ${}r$ define a unique point

${}P=(x,y)=(r\cos \alpha ,r\sin \alpha )=r(\cos \alpha ,\sin \alpha )\,$

in the real plane ${}\mathbb {R} ^{2}$. Here, ${}r$ is the distance between the point ${}P$ and the zero point ${}(0,0)$ and ${}(\cos \alpha ,\sin \alpha )$ means the intersecting point of the ray through ${}P$ with the unit circle. Every point ${}P=(x,y)\neq 0$ has a unique representation with ${}r={\sqrt {x^{2}+y^{2}}}$ and with an angle ${}\alpha$, which has to be chosen accordingly (the zero point is represented by ${}r=0$ and an arbitrary angle). The components ${}(r,\alpha )$ are called the polar coordinates of ${}P$.

Example

Every complex number ${}z\in \mathbb {C}$, ${}z\neq 0$, can be written uniquely as

${}z=r(\cos \alpha ,\sin \alpha )=(r\cos \alpha ,r\sin \alpha )=r\cos \alpha +(r\sin \alpha ){\mathrm {i} }\,$

with a positive real number ${}r$, which is the distance between ${}z$ and the zero point (thus, ${}r=\vert {z}\vert$) and an angle ${}\alpha$ between ${}0$ and below ${}360$ degree, measured counterclockwise starting with the positive real axis. The pair ${}(r,\alpha )$ constitutes the polar coordinates of the complex number.

Polar coordinates in the real plane and for complex numbers are the same. However, the polar coordinates allow a new interpretation of the multiplication of complex numbers: Because of

${}{\begin{aligned}(r\cos \alpha +{\mathrm {i} }r\sin \alpha )\cdot (s\cos \beta +{\mathrm {i} }s\sin \beta )&=rs(\cos \alpha \cos \beta -\sin \alpha \sin \beta )+{\mathrm {i} }rs(\cos \alpha \sin \beta +\sin \alpha \cos \beta )\\&=rs(\cos(\alpha +\beta )+{\mathrm {i} }\sin(\alpha +\beta ))\end{aligned}}$

(where we have used the addition theorems for sine and cosine), one can multiply two complex numbers by multiplying their modulus and adding their angles.

This new way of looking at the multiplication of complex numbers, yields also a new understanding of roots of complex numbers, which exist, due to the fundamental theorem of algebra. If ${}z=r\cos \alpha +r{\mathrm {i} }\sin \alpha$, then

${}w={\sqrt[{n}]{r}}\cos {\frac {\alpha }{n}}+{\sqrt[{n}]{r}}{\mathrm {i} }\sin {\frac {\alpha }{n}}\,$

is an ${}n$-th root of ${}z$. This means that one has to take the real ${}n$-th root of the modulus of the complex number and one has to divide the angle by ${}n$.

Example

A spatial variant of the polar coordinates are the so-called cylindrical coordinates. A triple ${}(r,\alpha ,z)\in \mathbb {R} _{+}\times [0,2\pi [\times \mathbb {R}$ is sent to the Cartesian coordinates

${}(x,y,z)=(r\cos \alpha ,r\sin \alpha ,z)\,.$