Parametric Equations; Polar Equations

9.4 Assess Your Understanding
Printed Page 668

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Concepts and Vocabulary

In a polar coordinate system, the origin is called the ______________, and the ______________ coincides with the positive $x$ -axis of the rectangular coordinate system.

Pole, polar axis

True or False Another representation in polar coordinates for the point $\left( 2,\dfrac{\pi }{3}\right)$ is $\left( 2,\dfrac{4\pi }{3}\right) .$

False

True or False In a polar coordinate system, each point in the plane has exactly one pair of polar coordinates.

False

True or False In a rectangular coordinate system, each point in the plane has exactly one pair of rectangular coordinates.

True

True or False In polar coordinates $( r,\theta)$ , the number $r$ can be negative.

True

True or False If $( r,\theta )$ are the polar coordinates of the point $P,$ then $\vert r\vert$ is the distance of the point $P$ from the pole.

True

To convert the point $( r,\theta )$ in polar coordinates to a point $( x,y)$ in rectangular coordinates, use the formulas $x=$ ______________ and $y=$ ______________.

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

An equation whose variables are polar coordinates is called a(n) ____________________________ .

Polar equation

Skill Building

In Problems 9–16, match each point in polar coordinates with A, B, C, or D on the graph.

$\left( 2,-\dfrac{11\pi }{6}\right)$

$\left(-2,-\dfrac{\pi }{6}\right)$

$\left( -2,\dfrac{\pi }{6}\right)$

$\left( 2,\dfrac{7\pi }{6}\right)$

$\left( 2,\dfrac{5\pi }{6}\right)$

$\left(-2,\dfrac{5\pi }{6}\right)$

$\left( -2,\dfrac{7\pi }{6} \right)$

$\left( 2,\dfrac{11\pi }{6}\right)$

In Problems 17–24, polar coordinates of a point are given. Plot each point in a polar coordinate system.

$\left( 4,\dfrac{\pi }{3}\right)$

$\left( -4,\dfrac{\pi }{3}\right)$

$\left( -4,-\dfrac{\pi }{3}\right)$

$\left( 4,-\dfrac{\pi }{3}\right)$

$\left( \sqrt{2},\dfrac{\pi }{4}\right)$

$\left( 7,-\dfrac{7\pi }{4}\right)$

$\left( -6,\dfrac{4\pi }{3}\right)$

$\left( 5,\dfrac{\pi }{2}\right)$

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In Problems 25–32, polar coordinates of a point are given. Find other polar coordinates $( r,\theta )$ of the point for which:

(a) $r>0,\;-2\pi \leq \theta <0$

(b) $r<0,\;0\leq \theta <2\pi$

(c) $r>0,\;2\pi \leq \theta <4\pi$

$\left( 5,\dfrac{2\pi }{3}\right)$

$\left(5, -\dfrac{4\pi}{3}\right)$
$\left(-5, \dfrac{5\pi}{3}\right)$
$\left(5, \dfrac{8\pi}{3}\right)$

$\left( 4,\dfrac{3\pi }{4}\right)$

$( -2,3\pi )$

$(2, -2\pi)$
$(-2, \pi)$
$(2, 2\pi)$

$(-3,4\pi )$

$\left( 1,\dfrac{\pi }{2}\right)$

$\left(1, -\dfrac{3\pi}{2}\right)$
$\left(-1, \dfrac{3\pi}{2}\right)$
$\left(1, \dfrac{5\pi}{2}\right)$

$( 2,\pi )$

$\left( -3,-\dfrac{\pi }{4}\right)$

$\left(3, -\dfrac{5\pi}{4}\right)$
$\left(-3, \dfrac{7\pi}{4}\right)$
$\left(3, \dfrac{11\pi}{4}\right)$

$\left( -2,-\dfrac{2\pi }{3}\right)$

In Problems 33–40, polar coordinates of a point are given. Find the rectangular coordinates of each point.

$\left( 6,\dfrac{\pi }{6}\right)$

$(3 \sqrt{3}, 3)$

$\left( -6,\dfrac{\pi }{6}\right)$

$\left( -6,-\dfrac{\pi }{6}\right)$

$(-3 \sqrt{3}, 3)$

$\left( 6,-\dfrac{\pi }{6}\right)$

$\left( 5,\dfrac{\pi }{2}\right)$

$(0, 5)$

$\left( 8,\dfrac{\pi }{4}\right)$

$\left( 2\sqrt{2},-\dfrac{\pi }{4}\right)$

$(2, -2)$

$\left( -5,-\dfrac{\pi }{3}\right)$

In Problems 41–50, rectangular coordinates of a point are given. Plot the point. Find polar coordinates $( r,\theta )$ of each point for which $r>0$ and $0\leq \theta <2\pi$ .

$( 5,0)$

$( 2,-2)$

$(-2,2)$

$( -2,-2\sqrt{3})$

$( \sqrt{3},1)$

$(0,-3)$

$( -\sqrt{3},1)$

$( 3\sqrt{2},-3\sqrt{2})$

$( 3,2)$

$( -6.5,1.2)$

In Problems 51–58, match each of the graphs (A) through (H) to one of the following polar equations.

$r=2$

$\theta = \dfrac{\pi }{4}$

$r=2\cos \theta$

$r\cos \theta =2$

$r=-2\cos \theta$

$r=2\sin \theta$

$\theta =\dfrac{3\pi }{4}$

$r\sin \theta =2$

In Problems 59–74, identify and graph each polar equation. Convert to a rectangular equation if necessary.

$r=4$

The circle centered at (0, 0) of radius 4.

$r=2$

$\theta =\dfrac{\pi }{3}$

A line through the origin.

$\theta =-\dfrac{\pi }{4}$

$r\sin \theta =4$

The line $y=4$ .

$r\cos \theta =4$

$r\cos \theta =-2$

The line $x=-2$ .

$r\sin \theta =-2$

$r=2\cos \theta$

The circle centered at (1, 0) of radius 1.

$r=2\sin \theta$

$r=-4\sin \theta$

The circle centered at (0, -2) of radius 2.

$r=-4\cos \theta$

$r\sec \theta =4$

The circle centered at (2, 0) of radius 2 excluding the pole.

$r\csc \theta =8$

$r\csc \theta =-2$

The circle centered at (0, -1) of radius 1, excluding the pole.

$r\sec \theta =-4$

In Problems 75–82, the letters $x$ and $y$ represent rectangular coordinates. Write each equation in polar coordinates $r$ and $\theta$ .

$\dfrac{x^{2}}{4}+\dfrac{y^{2}}{9}=1$

$r=6\dfrac{\sqrt{9\ \cos^2\theta + 4\ \sin^2\theta}}{9\ \cos^2\theta + 4\ \sin^2\theta}$

$x-4y+4=0$

$x^{2}+y^{2}-4x=0$

$r=4\ \cos\theta$

$y=-6$

$x^{2}=1-4y$

$r^2\ \cos^2\theta + 4r\ \sin\theta - 1 = 0$

$y^{2}=1-4x$

$xy=1$

$r=\dfrac{\sqrt{\cos\theta\ \sin\theta}}{\cos\theta \ \sin\theta}$

$x^{2}+y^{2}-2x+4y=0$

In Problems 83–94, the letters $r$ and $\theta$ represent polar coordinates. Write each equation in rectangular coordinates $x$ and $y$ .

$r=\cos \theta$

$\left(x-\dfrac{1}{2}\right)^2+y^2= \dfrac{1}{4}$

$r=2+\cos \theta$

$r^{2}=\sin \theta$

$y=(x^2+y^2)^{{3}/{2}}$

$r^{2}=1-\sin \theta$

$r=\dfrac{4}{1-\cos \theta }$

$\sqrt{x^2 + y^2} - x = 4$

$r=\dfrac{3}{3-\cos \theta }$

$r^{2}=\theta$

$y=x\ \tan(x^2+y^2)$

$\theta =-\dfrac{\pi }{4}$

$r=2$

$y=\sqrt{4-x^2}$

$r=-5$

$\tan \theta =4$

$y=4x$

$\cot \theta =3$

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Applications and Extensions

Chicago In Chicago, the road system is based on a rectangular coordinate system, with the intersection of Madison and State Streets at the origin, and east as the positive $x$ -axis. Intersections are indicated by the number of blocks they are from the origin. For example, Wrigley Field is located at 1060 West Addison, which is 10 blocks west of State Street and 36 blocks north of Madison Street.
1. (a) Write the location of Wrigley Field using rectangular coordinates.
2. (b) Write the location of Wrigley Field using polar coordinates. Use east as the polar axis.
3. (c) U.S. Cellular Field is located at 35th Street and Princeton, which is 3 blocks west of State Street and 35 blocks south of Madison Street. Write the location of U.S. Cellular Field using rectangular coordinates.
4. (d) Write the location of U.S. Cellular Field using polar coordinates.

$x=-10, y=36$
$(37.363, 1.842)$
$x=-3, y=-35$
$(35.128, 4.627)$

Show that the formula for the distance $d$ between two points $P_{1}=(r_{1},~\theta _{1})$ and $P_{2}=(r_{2},~\theta _{2})$ is $\begin{equation*} d=\sqrt{r_{1}^{2}+r_{2}^{2}-2r_{1}r_{2}\cos (\theta _{2}-\theta _{1})} \end{equation*}$

Horizontal Line Show that the graph of the equation $r\sin \theta =a$ is a horizontal line $a$ units above the pole if $a>0$ , and $|a|$ units below the pole if $a<0$ .

See Student Solutions Manual.

Vertical Line Show that the graph of the equation $\ r\cos \theta =a$ is a vertical line $a$ units to the right of the pole if $a>0$ , and $|a|$ units to the left of the pole if $a<0$ .

Circle Show that the graph of the equation $r=2a\sin \theta,\;a>0$ , is a circle of radius $a$ with its center at the rectangular coordinates $(0, a)$ .

See Student Solutions Manual.

Circle Show that the graph of the equation $r=-2a\sin \theta ,$ $a>0$ , is a circle of radius $a$ with its center at the rectangular coordinates $(0,\,-a)$ .

Circle Show that the graph of the equation $r=2a\cos \theta ,$ $a>0$ , is a circle of radius $a$ with its center at the rectangular coordinates $(a, 0)$ .

See Student Solutions Manual.

Circle Show that the graph of the equation $r=-2a\cos \theta ,a>0$ , is a circle of radius $a$ with its center at the rectangular coordinates $(-a, 0)$ .

Exploring Using Graphing Technology
1. Use a square screen to graph $r_{1}=\sin \theta ,$ $r_{2}=2\sin \theta$ , and $r_{3}=3\sin \theta .$
2. Describe how varying the constant $a,$ $a>0,$ alters the graph of $r=a\sin \theta$ .
3. Graph $r_{1}=-\sin \theta ,$ $r_{2}=-2\sin \theta ,$ and $r_{3}=-3\sin \theta .$
4. Describe how varying the constant $a,$ $a<0,$ alters the graph of $r=a\sin \theta$ .

(a)
(b) Answers will vary.
(c)
(d) Answers will vary.

Exploring Using Graphing Technology
1. Use a square screen to graph $r_{1}=\cos \theta ,$ $r_{2}=2\cos \theta ,$ and $r_{3}=3\cos \theta .$
2. Describe how varying the constant $a,$ $a>0,$ alters the graph of $r=a\cos \theta$ .
3. Graph $r_{1}=-\cos \theta ,$ $r_{2}=-2\cos \theta ,$ and $r_{3}=-3\cos \theta .$
4. Describe how varying the constant $a,$ $a<0,$ alters the graph of $r=a\cos \theta$ .

Challenge Problems

Show that $r=a\sin \theta +b\cos \theta ,$ $a,\,b$ not both zero, is the equation of a circle. Find the center and radius of the circle.

See Student Solutions Manual.

Express $r^{2}=\cos ( 2\theta )$ in rectangular coordinates free of radicals.

Prove that the area of the triangle with vertices $(0,0),~(r_{1},\theta _{1}),(r_{2},\theta _{2})$ is $\begin{equation*} A=\dfrac{1}{2}r_{1}r_{2}~\sin (\theta _{2}-\theta _{1})\qquad 0\leq \theta _{1}<\theta _{2}\leq \pi \end{equation*}$

See Student Solutions Manual.

9.4 Assess Your UnderstandingPrinted Page 668

9.4 Assess Your Understanding
Printed Page 668