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Modern Analytic Geometry

William Wooton, Edwin F. Beckenbach, Frank J. Fleming

Chapter 7

Polar Coordinates - all with Video Answers

Educators

Section 1

Polar Coordinate System

00:28

Problem 1

In Exercises 1-12, Graph the point with the given polar coordinates, and specify three additional pairs of polar coordinates for the point.
$\left(3,25^{\circ}\right)$

Robert Leedy
Robert Leedy
Numerade Educator

Problem 2

Graph the point with the given polar coordinates, and specify three additional pairs of polar coordinates for the point.
$\left(4,45^{\circ}\right)$

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Problem 3

Graph the point with the given polar coordinates, and specify three additional pairs of polar coordinates for the point.
$\left(2,120^{\circ}\right)$

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01:00

Problem 4

Graph the point with the given polar coordinates, and specify three additional pairs of polar coordinates for the point.
$\left(3,150^{\circ}\right)$

Dharmendra Jain
Dharmendra Jain
Numerade Educator

Problem 5

Graph the point with the given polar coordinates, and specify three additional pairs of polar coordinates for the point.
$\left(2, \frac{\pi^R}{2}\right)$

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Problem 6

Graph the point with the given polar coordinates, and specify three additional pairs of polar coordinates for the point.
$\left(3, \frac{3 \pi^R}{4}\right)$

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01:24

Problem 7

Graph the point with the given polar coordinates, and specify three additional pairs of polar coordinates for the point.
$\left(1, \frac{7 \pi^k}{6}\right)$

Carson Merrill
Carson Merrill
Numerade Educator
01:24

Problem 8

Graph the point with the given polar coordinates, and specify three additional pairs of polar coordinates for the point.
$\left(4, \frac{7 \pi^R}{4}\right)$

Carson Merrill
Carson Merrill
Numerade Educator

Problem 9

Graph the point with the given polar coordinates, and specify three additional pairs of polar coordinates for the point.
$\left(-1,40^{\circ}\right)$

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00:47

Problem 10

Graph the point with the given polar coordinates, and specify three additional pairs of polar coordinates for the point.
$\left(-3,100^{\circ}\right)$

Linh Vu
Linh Vu
Numerade Educator
01:33

Problem 11

Graph the point with the given polar coordinates, and specify three additional pairs of polar coordinates for the point.
$\left(-2, \frac{5 \pi^R}{6}\right)$

Monica Miller
Monica Miller
Numerade Educator
01:12

Problem 12

Graph the point with the given polar coordinates, and specify three additional pairs of polar coordinates for the point.
$\left(-4, \frac{11 \pi^k}{6}\right)$

Monica Miller
Monica Miller
Numerade Educator
00:24

Problem 13

In Exercises 13-24, Find the Cartesian coordinates of the point with the given polar coordinates.
$\left(2,45^{\circ}\right)$

Yujie Wang
Yujie Wang
College of San Mateo
00:47

Problem 14

Find the Cartesian coordinates of the point with the given polar coordinates.
$\left(3,30^{\circ}\right)$

Linh Vu
Linh Vu
Numerade Educator
01:16

Problem 15

Find the Cartesian coordinates of the point with the given polar coordinates.
$\left(6,90^{\circ}\right)$

Monica Miller
Monica Miller
Numerade Educator
02:23

Problem 16

Find the Cartesian coordinates of the point with the given polar coordinates.
$\left(5,270^{\circ}\right)$

Heather Zimmers
Heather Zimmers
Numerade Educator
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Problem 17

Find the Cartesian coordinates of the point with the given polar coordinates.
$\left(3,180^{\circ}\right)$

AG
Ankit Gupta
Numerade Educator
02:08

Problem 18

Find the Cartesian coordinates of the point with the given polar coordinates.
$\left(4,360^{\circ}\right)$

Taylor Shimono
Taylor Shimono
Numerade Educator
01:53

Problem 19

Find the Cartesian coordinates of the point with the given polar coordinates.
$\left(2,135^{\circ}\right)$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
02:23

Problem 20

Find the Cartesian coordinates of the point with the given polar coordinates.
$\left(3,240^{\circ}\right)$

Heather Zimmers
Heather Zimmers
Numerade Educator
00:57

Problem 21

Find the Cartesian coordinates of the point with the given polar coordinates.
$\left(-2,330^{\circ}\right)$

Monica Miller
Monica Miller
Numerade Educator
00:50

Problem 22

Find the Cartesian coordinates of the point with the given polar coordinates.
$\left(-1,60^{\circ}\right)$

Aamir Mithaiwala
Aamir Mithaiwala
Numerade Educator
00:57

Problem 23

Find the Cartesian coordinates of the point with the given polar coordinates.
$\left(-3,150^{\circ}\right)$

Yujie Wang
Yujie Wang
College of San Mateo
00:43

Problem 24

Find the Cartesian coordinates of the point with the given polar coordinates.
$\left(-5,225^{\circ}\right)=$

Linda Hand
Linda Hand
Numerade Educator
01:04

Problem 25

In Exercises 25-36, Find two pairs of polar coordinates, one pair with $r>0$ and the other pair with $r<0$, for the point whose Cartesian coordinates are given. In each case, choose $\theta$ so that $0 \leq m^{\circ}(\theta)<360$.
$(\sqrt{3},-3)$

Tyler Moulton
Tyler Moulton
Numerade Educator
01:04

Problem 26

Find two pairs of polar coordinates, one pair with $r>0$ and the other pair with $r<0$, for the point whose Cartesian coordinates are given. In each case, choose $\theta$ so that $0 \leq m^{\circ}(\theta)<360$.
$(4,4)$

Carson Merrill
Carson Merrill
Numerade Educator

Problem 27

Find two pairs of polar coordinates, one pair with $r>0$ and the other pair with $r<0$, for the point whose Cartesian coordinates are given. In each case, choose $\theta$ so that $0 \leq m^{\circ}(\theta)<360$.
$(0,5)$

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01:53

Problem 28

Find two pairs of polar coordinates, one pair with $r>0$ and the other pair with $r<0$, for the point whose Cartesian coordinates are given. In each case, choose $\theta$ so that $0 \leq m^{\circ}(\theta)<360$.
$(2 \sqrt{3},-2)$

Monica Miller
Monica Miller
Numerade Educator
01:24

Problem 29

Find two pairs of polar coordinates, one pair with $r>0$ and the other pair with $r<0$, for the point whose Cartesian coordinates are given. In each case, choose $\theta$ so that $0 \leq m^{\circ}(\theta)<360$.
$(3,-3)$

Linh Vu
Linh Vu
Numerade Educator
05:30

Problem 30

Find two pairs of polar coordinates, one pair with $r>0$ and the other pair with $r<0$, for the point whose Cartesian coordinates are given. In each case, choose $\theta$ so that $0 \leq m^{\circ}(\theta)<360$.
$(-3,0)$

Sonam Khatri
Sonam Khatri
Numerade Educator
01:11

Problem 31

Find two pairs of polar coordinates, one pair with $r>0$ and the other pair with $r<0$, for the point whose Cartesian coordinates are given. In each case, choose $\theta$ so that $0 \leq m^{\circ}(\theta)<360$.
$(-4,-4 \sqrt{3})$

Linh Vu
Linh Vu
Numerade Educator

Problem 32

Find two pairs of polar coordinates, one pair with $r>0$ and the other pair with $r<0$, for the point whose Cartesian coordinates are given. In each case, choose $\theta$ so that $0 \leq m^{\circ}(\theta)<360$.
$(-5,-5)$

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01:11

Problem 33

Find two pairs of polar coordinates, one pair with $r>0$ and the other pair with $r<0$, for the point whose Cartesian coordinates are given. In each case, choose $\theta$ so that $0 \leq m^{\circ}(\theta)<360$.
$(0,-4)$

Linh Vu
Linh Vu
Numerade Educator
01:10

Problem 34

Find two pairs of polar coordinates, one pair with $r>0$ and the other pair with $r<0$, for the point whose Cartesian coordinates are given. In each case, choose $\theta$ so that $0 \leq m^{\circ}(\theta)<360$.
$(-\sqrt{3}, 1)$

Linh Vu
Linh Vu
Numerade Educator
00:42

Problem 35

Find two pairs of polar coordinates, one pair with $r>0$ and the other pair with $r<0$, for the point whose Cartesian coordinates are given. In each case, choose $\theta$ so that $0 \leq m^{\circ}(\theta)<360$.
$(2,-2)$

Linh Vu
Linh Vu
Numerade Educator
View

Problem 36

Find two pairs of polar coordinates, one pair with $r>0$ and the other pair with $r<0$, for the point whose Cartesian coordinates are given. In each case, choose $\theta$ so that $0 \leq m^{\circ}(\theta)<360$.
$(6,-2 \sqrt{3})$

Carson Merrill
Carson Merrill
Numerade Educator
01:37

Problem 37

Prove that if a point $\mathbf{S}\left(x_1, y_1\right)$ has polar coordinates $(r, \theta)$, and a point $\mathbf{T}\left(x_2, y_2\right)$ has polar coordinates $(-r, \theta)$, then $x_1=-x_2$ and $y_1=-y_2$.

Linh Vu
Linh Vu
Numerade Educator
01:37

Problem 38

Use the fact that for all $\theta, \sin (-\theta)=-\sin \theta$ and $\cos (-\theta)=\cos \theta$ to prove that if a point $\mathbf{S}\left(x_1, y_1\right)$ has polar coordinates $(r, \theta)$ and a point $\mathbf{T}\left(x_2, y_2\right)$ has polar coordinates $(r,-\theta)$, then $x_1=x_2$, and $y_1=-y_2$.

Linh Vu
Linh Vu
Numerade Educator
01:37

Problem 39

Use the fact that for all $\theta, \sin \left(\pi^8-\theta\right)=\sin \theta$ and $\cos \left(\pi^8-\theta\right)=-\cos \theta$ to prove that is a point $\mathbf{S}\left(x_1, y_1\right)$ has polar coordinates $(r, \theta)$ and a point $\mathbf{T}\left(x_2, y_2\right)$ has polar coordinates $\left(r, \pi^{\mathrm{R}}-\theta\right)$ then $x_1=-x_2$ and $y_1=y_2$.

Linh Vu
Linh Vu
Numerade Educator
01:37

Problem 40

Use the fact that for all $\theta, \sin \left(\frac{\pi^R}{2}-\theta\right)=\cos \theta$ and $\cos \left(\frac{\pi^R}{2}-\theta\right)=\sin \theta$ to prove that if a point $\mathbf{S}\left(x_1, y_1\right)$ has polar coordinates $(r, \theta)$ and a point $\mathbf{T}\left(x_2, y_2\right)$ has polar coordinates $\left(r, \frac{\pi^R}{2}-\theta\right)$, then $x_1=y_2$ and $y_1=x_2$.

Linh Vu
Linh Vu
Numerade Educator