Section 6.4a:

Our standard (x, y) system is called the_____________ coordinate systemrectangularrectangular

Today, we start learning aboutthe POLARPOLAR coordinate system…

Polar Coordinate Polar Coordinate SystemSystem

The polar coordinate system is a plane with point O,the pole, and a ray from O, the polar axis. Eachpoint P in the plane is assigned polar coordinates:

r is the directed distance from O to Pr is the directed distance from O to P

0 is the directed angle whose initial side0 is the directed angle whose initial side is on the polar axis and whose terminalis on the polar axis and whose terminal side is on the line OPside is on the line OP

The polar coordinate system is a plane with point O,the pole, and a ray from O, the polar axis. Eachpoint P in the plane is assigned polar coordinates:

r is the directed distance from O to Pr is the directed distance from O to P

0 is the directed angle whose initial side0 is the directed angle whose initial side is on the polar axis and whose terminalis on the polar axis and whose terminal side is on the line OPside is on the line OP

OPole

Polar axis

P(r, )

r

Practice ProblemsPractice Problems

πP 2,

3

3πQ 3,

4

Plot the points with the given polar coordinates.

R 1, 49

Practice ProblemsPractice Problems

πP 2,

4

Plot the points with the given polar coordinates.

Are there any Are there any other other coordinates thatcoordinates thatcould be given for point P? If so, howcould be given for point P? If so, howmany are there?many are there?

Finding all Polar Finding all Polar CoordinatesCoordinatesof a Pointof a Point

r,θ 2nπ or r,θ 2n 1 π

πP 2,

4

Let P have polar coordinates (r, ). Any other polarcoordinate of P must be of the form

where n is any integer. In particular, the pole haspolar coordinates (0, ), where is any angle.

So, what are all of the polar coordinates for ourSo, what are all of the polar coordinates for ourprevious example:previous example:

π π2, 2nπ or 2, 2n 1 π4 4

Coordinate ConversionCoordinate Conversion

cosθx rsin θy r

2 2 2r x y

Look at a point P represented graphically in bothpolar and rectangular coordinates:

x

y

r

Polar axisx

y

Pole

O(0,0)

P(r, 0)P(x, y)

0

tan θy

x

More Practice ProblemsMore Practice Problems

5πC 3,

6

Find the rectangular coordinates of the points with thegiven polar coordinates.

3 3 3C ,

2 2

Use a graph to supportUse a graph to supportyour answers???your answers???

More Practice ProblemsMore Practice Problems

K 2, 200

Find the rectangular coordinates of the points with thegiven polar coordinates.

K 1.879,0.684

Use a graph to supportUse a graph to supportyour answers???your answers???

More Practice ProblemsMore Practice Problems

H( 1,1)

Find two polar coordinate pairs for the points withthe given rectangular coordinates.

π 3π2, and 2,

4 4

Use a graphUse a graphto supportto support

your answers?your answers?

More Practice ProblemsMore Practice Problems

D( 3,0)

Find two polar coordinate pairs for the points withthe given rectangular coordinates.

3,0 and 3, π

Use a graphUse a graphto supportto support

your answers?your answers?

More Practice ProblemsMore Practice ProblemsPolar coordinates of point P are given. Find all of itspolar coordinates.

1, 4P

and1, 24

n

1, 2 1

4n

(n an integer)

More Practice ProblemsMore Practice ProblemsRectangular coordinates of point P are given. Find allpolar coordinates of P that satisfy:

0 2 (a)

1,3P (b) 0 4 (c)

10,1.249(a) or 10,4.391

10,1.249(b) or 10, 1.893

More Practice ProblemsMore Practice ProblemsRectangular coordinates of point P are given. Find allpolar coordinates of P that satisfy:

0 2 (a)

1,3P (b) 0 4 (c)

10,1.249(c) or 10,4.391

10,7.532 or 10,10.674or