Polar Coordinates

In the polar coordinate system the location of a point \(P\) in the plane is determined by an ordered pair \((r, \theta)\), where \(r\) is the distance from the pole \(O\) to \(P\) and \(\theta\) is the angle formed by the polar axis and the segment \(\overrightarrow{O P}\), as shown in the figure.

Polar and Rectangular Coordinates

Any point \(P\) in the plane has polar coordinates \(P(r, \theta)\) and rectangular coordinates \(P(x, y)\), as shown.

\[ x=r \cos \theta \quad \text { and } \quad y=r \sin \theta \]
\[ r^{2}=x^{2}+y^{2} \quad \text { and } \quad \tan \theta=\frac{y}{x} \]

Polar Equations and Graphs

A polar equation is an equation in the variables \(r\) and \(\theta\). The graph of a polar equation \(r=f(\theta)\) consists of all points \((r, \theta)\) whose coordinates satisfy the equation.

Symmetry in Graphs of Polar Equations

The graph of a polar equation is

Complex Numbers

A complex number is a number of the form \(a+b i\), where \(i^{2}=-1\) and where \(a\) and \(b\) are real numbers. For the complex number \(z=a+b i, a\) is called the real part and \(b\) is called the imaginary part. A complex number \(a+b i\) is graphed in the complex plane as shown.

The modulus (or absolute value) of a complex number \(z=a+b i\) is

\[ |z|=\sqrt{a^{2}+b^{2}} \]

Polar Form of Complex Numbers

A complex number \(z=a+b i\) has the polar form (or trigonometric form)

\[ z=r(\cos \theta+i \sin \theta) \]

where \(r=|z|\) and \(\tan \theta=b / a\). The number \(r\) is the modulus of \(z\) and \(\theta\) is the argument of \(z\).

Multiplication and Division of Complex Numbers in Polar Form

Suppose the complex numbers \(z_{1}\) and \(z_{2}\) have the following polar form:

\[ \begin{aligned} & z_{1}=r_{1}\left(\cos \theta_{1}+i \sin \theta_{1}\right) \\ & z_{2}=r_{2}\left(\cos \theta_{2}+i \sin \theta_{2}\right) \end{aligned} \]

Then

\[ \begin{aligned} z_{1} z_{2} & =r_{1} r_{2}\left[\cos \left(\theta_{1}+\theta_{2}\right)+i \sin \left(\theta_{1}+\theta_{2}\right)\right] \\ \frac{z_{1}}{z_{2}} & =\frac{r_{1}}{r_{2}}\left[\cos \left(\theta_{1}-\theta_{2}\right)+i \sin \left(\theta_{1}-\theta_{2}\right)\right] \end{aligned} \]

De Moivre's Theorem

If \(z=r(\cos \theta+i \sin \theta)\) is a complex number in polar form and \(n\) is a positive integer, then

\[ z^{n}=r^{n}(\cos n \theta+i \sin n \theta) \]

nth Roots of Complex Numbers

If \(z=r(\cos \theta+i \sin \theta)\) is a complex number in polar form and \(n\) is a positive integer, then \(z\) has the \(n\) distinct \(n\)th roots \(w_{0}, w_{1}, \ldots, w_{n-1}\), where

\[ w_{k}=r^{1 / n}\left[\cos \left(\frac{\theta+2 k \pi}{n}\right)+i \sin \left(\frac{\theta+2 k \pi}{n}\right)\right] \]

where \(k=0,1,2, \ldots, n-1\).

Finding the \(n\)th Roots of \(z\)

To find the \(n\)th roots of \(z=r(\cos \theta+i \sin \theta)\), we use the following observations:

  1. The modulus of each \(n\)th root is \(r^{1 / n}\).
  2. The argument of the first root \(w_{0}\) is \(\theta / n\).
  3. Repeatedly add \(2 \pi / n\) to get the argument of each successive root.

Parametric Equations

If \(f\) and \(g\) are functions defined on an interval \(I\), then the set of points \((f(t), g(t))\) is a plane curve. The equations

\[ x=f(t) \quad y=g(t) \]

where \(t \in I\), are parametric equations for the curve, with parameter \(t\).

Polar Equations in Parametric Form

The graph of the polar equation \(r=f(\theta)\) is the same as the graph of the parametric equations

\[ x=f(t) \cos t \quad y=f(t) \sin t \]

Vectors

A vector is a quantity with both magnitude and direction. A vector in the coordinate plane is expressed in terms of two coordinates or components

\[ \mathbf{v}=\left\langle a_{1}, a_{2}\right\rangle \]

If a vector \(\mathbf{v}\) has its initial point at \(P\left(x_{1}, y_{1}\right)\) and its terminal point at \(Q\left(x_{2}, y_{2}\right)\), then

\[ \mathbf{v}=\left\langle x_{2}-x_{1}, y_{2}-y_{1}\right\rangle \]

Let \(\mathbf{u}=\left\langle a_{1}, a_{2}\right\rangle, \mathbf{v}=\left\langle b_{1}, b_{2}\right\rangle\), and \(c \in \mathbb{R}\). The operations on vectors are defined as follows.

\[ \begin{aligned} \mathbf{u}+\mathbf{v} & =\left\langle a_{1}+b_{1}, a_{2}+b_{2}\right\rangle & & \text { Addition } \\ \mathbf{u}-\mathbf{v} & =\left\langle a_{1}-b_{1}, a_{2}-b_{2}\right\rangle & & \text { Subtraction } \\ c \mathbf{u} & =\left\langle c a_{1}, c a_{2}\right\rangle & & \text { Scalar multiplication } \end{aligned} \]

The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are defined by

\[ \mathbf{i}=\langle 1,0\rangle \quad \mathbf{j}=\langle 0,1\rangle \]

Any vector \(\mathbf{v}=\left\langle a_{1}, a_{2}\right\rangle\) can be expressed as

\[ \mathbf{v}=a_{1} \mathbf{i}+a_{2} \mathbf{j} \]