copyright © 2011 pearson, inc. 6.4 polar coordinates

Copyright © 2011 Pearson, Inc.

6.4Polar

Coordinates

4 - 2 Copyright © 2011 Pearson, Inc.

What you’ll learn about

Polar Coordinate System Coordinate Conversion Equation Conversion Finding Distance Using Polar Coordinates

… and whyUse of polar coordinates sometimes simplifies complicated rectangular equations and they are useful in calculus.


The Polar Coordinate System

A polar coordinate system is a plane with a point O, the pole, and a ray from O, the polar axis, as shown. Each point P in the plane is assigned polar coordinates as follows: r is the directed distance from O to P, and is the directed angle whose initial side is on the polar axis and whose terminal side is on the line OP.


Example Plotting Points in the Polar Coordinate System

Plot the points with the given polar coordinates.

(a) P(2, π / 3) (b) Q(−1, 3π / 4) (c) R(3, −45o)


Finding all Polar Coordinates of a Point

Let the point P have polar coordinates (r,). Any

other polar coordinate of P must be of the form

(r, + 2πn) or (−r, + (2n+1)π )where n is any integer. In particular, the pole has

polar coordinates (0,), where is any angle.


Coordinate Conversion Equations

Let the point P have polar coordinates (r,)and rectangular coordinates (x, y). Then

x=rcosy=rsinr2 =x2 + y2

tan =yx.


Example Converting from Polar to Rectangular Coordinates

Find the rectangular coordinates of the point with the

polar coordinates (2, 7π / 6).


Example Converting from Polar to Rectangular Coordinates

For point (2,7π / 6), r =2 and =7π / 6.x=rcos y=rsinx=2cos7π / 6 y=2sin7π / 6

x=2 −32

⎛

⎝⎜

⎞

⎠⎟ y=2 −

12

⎛

⎝⎜⎞

⎠⎟

x=− 3 y=−1

The recangular coordinate is − 3,−1( ).

Find the rectangular coordinates of the point with the

polar coordinates (2, 7π / 6).


Example Converting from Rectangular to Polar Coordinates

Find two polar coordinate pairs for the point with

the rectangular coordinates (1,−1).


Example Converting from Rectangular to Polar Coordinates

For the point (1,−1), x=1 and y=−1.

r2 =x2 + y2 tan =yx

r2 =12 + (−1)2 tan =−11

r2 = 2 tan =−1

r =± 2 =−π4+ nπ

Find two polar coordinate pairs for the point with

the rectangular coordinates (1,−1).

Two polar

coordinates

pairs are

2,−π4

⎛

⎝⎜⎞

⎠⎟ and

− 2,3π4

⎛

⎝⎜⎞

⎠⎟.


Example Converting from Polar Form to Rectangular Form

Convert r =−2csc to rectangular form and

identify the graph.Support your answer with a polar graphing utility.



r =−2cscr

csc=−2

rsin =−2y=−2

Convert r =−2csc to rectangular form and

identify the graph.Support your answer with a polar graphing utility.

The graph is the

horizontal line

y =−2.



Convert x −2( )2+ y−3( )

2=13 to polar form.



x −2( )2+ y−3( )

2=13

x2 −4x+ 4 + y2 −6y+9 =13

x2 + y2 −4x−6y=0

Substitute r2 =x2 + y2 ,x=rcos, and y=rsin.

Convert x −2( )2+ y−3( )

2=13 to polar form.



Convert x −2( )2+ y−3( )

2=13 to polar form.

r =0 is a single point that is also on the other graph.Thus the equation is r =4cos +6sin.

Substitute r 2 =x2 + y2 ,x=rcos, and y=rsin.r2 −4rcos −6rsin =0

r r −4cos −6sin( ) =0

r =0 or r =4cos +6sin


Quick Review

1. Determine the quadrants containing the terminal

side of the angle: −4π / 32. Find a positive and negative angle coterminal

with the given angle: −π / 33. Write a standard form equation for the circle with

center at ( −6,0) and a radius of 4.


Quick Review

Use the Law of Cosines to find the measure of the third side of the given triangle.

4.40º

8 10

5.

35º

6 11


Quick Review Solutions

1. Determine the quadrants containing the terminal

side of the angle: −4π / 3 II2. Find a positive and negative angle coterminal

with the given angle: −π / 3 5π /3, −7π /33. Write a standard form equation for the circle

with center at ( −6,0) and a radius of 4.

(x+6)2 + y2 =16


Quick Review Solutions

Use the Law of Cosines to find the measure of the third side of the given triangle.

4.40º

8 10

5.

35º

6 11

6.47

copyright © 2011 pearson, inc. 6.4 polar coordinates

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