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)
x4
6.Find a rectangular equation that has the same graph as the polar equation
4, / 3
a)
f)
b)
r7
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