7.7: Polar Coordinates

The position of a point in the plane can be described by its distance and direction from the origin. In measuring direction we take the $x$-axis as the starting point. Let $X$ be the point $(1,0)$ on the $x$-axis and let $P$ be a point in the plane as in Figure 7.7.1.

A pair of polar coordinates of $P$ is given by $(r, \theta)$ where $r$ is the distance from the origin to $P$ and $\theta$ is the angle $X O P$.

Each pair of real numbers $(r, \theta)$ determines a point $P$ in polar coordinates. To find $P$ we first rotate the line $O X$ through an angle $\theta$, forming a new line $O X^{\prime}$, and then go out a distance $r$ along the line $O X^{\prime}$. If $\theta$ is negative then the rotation is in the negative, or clockwise direction. If $r$ is negative the distance is measured along the line $O X^{\prime}$ in the direction away from $X^{\prime}$ (see Figure 7.7.2).

Figure 7.7.2

Example 1

Plot the following points in polar coordinates.

Solution

$$(2, \pi / 4), \quad(-1, \pi / 4), \quad(3,3 \pi / 4), \quad(2,-\pi / 4), \quad(-4,-\pi / 4) \nonumber$$

The solution is shown in Figure 7.7.3.

Each point $P$ has infinitely many different polar coordinate pairs. We see in Figure 7.7.4 that the point $P(3, \pi / 2)$ has all the coordinates

$$\left.\begin{array}{l} (3, \pi / 2+2 n \pi) \\ (-3,3 \pi / 2+2 n \pi) \end{array}\right\} n \text { an integer. } \nonumber$$

Figure 7.7.3

Figure 7.7.4

Figure 7.7 .5

Any coordinate pair $(0, \theta)$ with $r=0$ determines the origin. As we see in Figure 7.7.5, the coordinates of a point $P$ in rectangular and in polar coordinates are related by the equations

$$x=r \cos \theta, \quad y=r \sin \theta \nonumber$$

The graph, or locus in polar coordinates of a system of formulas in the variables $r, \theta$ is the set of all points $P(r, \theta)$ for which the formulas are true.

Example 2

The graph of the equation $r=a$ is the circle of radius $a$ centered at the origin (Figure 7.7.6(a)). The graph of the equation $\theta=b$ is a straight line through the origin (Figure 7.7.6(b)).

Solution

EXAMPLE 3 The graph of the system of formulas

$$r=\theta, \quad 0 \leq \theta \nonumber$$

is the spiral of Archimedes formed by moving a pencil along the line $O X$ while the line is rotating, with the pencil moving at the same speed as the point $X$. The graph is shown in Figure 7.7.6(c).

An equation in rectangular coordinates can readily be transformed into an equation in polar coordinates with the same graph by using $x=r \cos \theta, y=r \sin 0$.

(a)

(b)

(c)

Here are the polar equations for various types of straight lines. Examples of their graphs are shown in Figure 7.7.7.

(1) Line through the origin (not vertical).

Rectangular equation: $y=m x$.

Polar equation: $\quad r \sin \theta=m r \cos \theta$,

or:

$\tan \theta=m$

(2) Horizontal line (not through origin).

Rectangular equation: $\quad y=b$.

Polar equation: $\quad r \sin \theta=b$,

or:

$$r=b \csc \theta \nonumber$$

(3) Vertical line (not through origin).

Rectangular equation: $\quad x=a$.

Polar equation: $\quad r \cos \theta=a$,

or:

$$r=a \sec \theta \nonumber$$

(4) Vertical line through origin.

Rectangular equation: $\quad x=0$.

Polar equation: $\quad r \cos \theta=0$,

or:

$$\theta=\pi / 2 \nonumber$$

(5) Other lines.

Rectangular equation: $\quad y=m x+b$.

Polar equation:

$$r \sin \theta=m r \cos \theta+b \nonumber$$

or:

$$r=\frac{b}{\sin \theta-m \cos \theta} \nonumber$$

$y=m x, \tan \theta=m$

$x=a, r=a \sec \theta$

$y=b, r=b \csc \theta$

$$y=m x+b, r=\frac{b}{\sin \theta-m \cos \theta} \nonumber$$

Example 4

The parabola $y=x^{2}$ has the polar equation

Solution

$$r \sin \theta=(r \cos \theta)^{2}, \text { or } \quad r=\frac{\sin \theta}{\cos ^{2} \theta}=\tan \theta \sec \theta \nonumber$$

Example 5

The curve $y=1 / x$ has the polar equation

Solution

$$r \sin \theta=\frac{1}{r \cos \theta}, \quad \text { or } \quad r^{2}=\sec \theta \csc \theta \nonumber$$

The graph is shown in Figure 7.7.8.

Figure 7.7.8

Some curves have much simpler equations in polar coordinates than in rectangular coordinates.

Example 6

The graph of the equation

Solution

$$r=a \sin \theta \nonumber$$

is the circle one of whose diameters is the line from the origin to a point $a$ above the origin.

This can be seen from Figure 7.7.9, if we remember that a diameter and a point on the circle form a right triangle.

As $\theta$ increases, the point ( $a \sin \theta, \theta$ ) goes around this circle once for every $\pi$ radians.

Figure 7.7.9

Figure 7.7.10

An equation $r=f(\theta)$ in polar coordinates has the same graph as the pair of parametric equations

$$x=f(\theta) \cos \theta, \quad y=f(\theta) \sin \theta \nonumber$$

in rectangular coordinates. This can be seen from Figure 7.7.10.

EXAMPLE 7

(a) The spiral $r=\theta$ has the parametric equations

$$x=\theta \cos \theta, \quad y=\theta \sin \theta \nonumber$$

(b) The circle $r=a \sin \theta$ has the parametric equations

$$x=a \sin \theta \cos \theta, \quad y=a \sin ^{2} \theta \nonumber$$

PROBLEMS FOR SECTION 7.7

1 Plot the following points in polar coordinates:
(a) $(2, \pi / 3)$
(b) $(-3, \pi / 2)$
(c) $(1,4 \pi / 3)$
(d) $(-2,-\pi / 4)$
(e) $\left(\frac{1}{2}, \pi\right)$
(f) $(0,3 \pi / 2)$

In Problems 2-12, find an equation in polar coordinates which has the same graph as the given equation in rectangular coordinates.

$$\begin{array}{rlrl} \mathbf{2} & y=3 x & \mathbf{3} & y=5 x+2 \\ \mathbf{4} & y=-4 & \mathbf{5} & x=2 \\ \mathbf{6} & x y^{2}=1 & \mathbf{7} & y=x^{2}+1 \\ \mathbf{8} & x^{2}+y^{2}=5 & \mathbf{9} & y=3 x^{2}-2 x \\ \mathbf{1 0} & y=x^{3} & \mathbf{1 1} & y=x^{2}+y^{2} \\ \mathbf{1 2} & y=\sin x & & \end{array} \nonumber$$

In Problems 13-20, sketch the given curve in polar coordinates.
13
$r=\cos \theta$
$14 r=-\sec \theta$
$15 r=\sin (\theta+\pi / 4)$
$16 \quad r=0, \quad 0 \leq 0$

$$\begin{array}{llll} 17 & r=1+\theta^{2} / \pi^{2} & 18 & r=\frac{1}{\sin \theta+\cos \theta} \\ 19 & r=\cot \theta \csc \theta & 20 & r^{2}=-2 \sec \theta \csc \theta \end{array} \nonumber$$

In Problems 21-24, find rectangular parametric equations for the given curves.

21	$r=\sin (3 \theta)$	22	$r=\sec \theta \csc \theta$
23	$r=\theta^{2}$	24	$r=\tan \theta$

25 Prove that if $f(\theta)=f(-\theta)$ then the curve $r=f(\theta)$ is symmetric about the $x$-axis. That is, if $(x, y)$ is on the curve then so is $(x,-y)$.

26 Prove that if $f(\theta)=f(\pi+\theta)$ then the curve $r=f(\theta)$ is symmetric about the origin. That is, if $(x, y)$ is on the curve so is $(-x,-y)$

27 Prove that if $f(\theta)=f(\pi-\theta)$ then the curve $r=f(\theta)$ is symmetric about the $y$-axis