8 1

8.1 Polar Coordinates   

Definition of polar coordinates Relationship between polar and rectangular coordinates Polar equations

Definition of Polar Coordinates

Point O-

pole/origin

Polar

axis-a ray from point O Point P can be assigned as

r

P (r ,  )

is the distance from O to P



is the angle between the

polar axis and segment

Example 1: Plotting points in polar coordinates Plot the points whose polar coordinates are given. a)

b)

 4,  / 6

c)

 2, / 4

  3,3 / 4

Equivalent Polar Points ,

P (r ,  ) P (r ,  2n )

and

P( r ,   (2n  1) )

Example 2: Different Polar coordinates for the Same Point a) Graph the point with polar coordinates

P( 2,  / 6)

.

b) Find two other polar coordinate representations of P with r>0 and two with r<0.

OP

Relationship Between Polar and Rectangular Coordinates Example 3: Polar to

Rectangular

 2,  / 6

Convert in polar coordinates to rectangular coordinates.

Example 4: Rectangular to Polar Convert

  1,1

in rectangular coordinates to polar coordinates.

Polar Equations Example 5: Rectangular equation to polar equation Find a polar equation that has the same graph as a) b)

x 2  y 2  8x x 2  8(2  y )

Example 6: Polar equation to rectangular equation

Find a rectangular equation that has the same graph as the polar equation

r 2  9 cos 2

.

Tutorial 8.1 1.Plot the point that has the given polar coordinates. a)

 4,  / 4

b)

c)

1,0

d)

 6,7 / 6

 3,  

e)

 1    , / 2  2 

f)

  4, / 6

2.Find two other polar coordinate representations of P with r>0 and two with r<0. a)

 3,  / 2

b)

c)

  5,0

  1,7 / 6

d)

e)

 2,3 / 4

3.Find rectangular coordinates for the point whose polar coordinates are given. a)

 4,  / 6

b)



2 , / 4

c)



d)

 5,5 

6

2 ,11 / 6

4.Find polar coordinates for the point whose rectangular coordinates are given. a)

  1,1

b)



8, 8



c)

 3,4

d)

  6,0

5.Find a polar equation that has the same graph as a)

x y

b)

y  x2

c)

x4

6.Find a rectangular equation that has the same graph as the polar equation



 4,  / 3

a)

f)

b)

r7

g)

r  1  2 sin 

c)

   2

d)

r cos   6

h)

1 r sin   cos 

e)

r  4 sin 

i)

4 r 1  2 sin 

r  1  cos 

r 2  tan 

Tutorial 8.1 : ANSWERS 2)a)

3)a)

4b) 4b) 5b) 6b) 6e) 6g) 6i)

3   5     3, ,  3,  2  2  

2

3 ,2



b)

b)

5        1, , 1,  6   6 

c)

1,1   5,0 4c)

d)

c)

(3 6 ,3 2 ) 4a)

4d) 5c) 6d)

6b)

  5,2  ,  5,  

6a) 6f)

6h) 6j)

5a)

d)

5   7    2, ,   2,  4  4  

e)

5   4    4, ,   4,  3 3   