Do Now - #18 on p.558
Graph the set of points whose polar coordinates satisfy the
given equations and inequalities.
0   2
1 r  2
1
2
Relating Polar and
Cartesian Coordinates
Section 10.5b
Relating Polar and Cartesian Coordinates
Ray


Coordinate Conversion
Equations:
P  x, y 
P  r , 
2
r
Common
origin
O
y
 0

x
Initial Ray
x  r cos 
y  r sin 
x y r
y
 tan 
x
2
2
2
Relating Polar and Cartesian Coordinates
Some curves are easier to work with in polar coordinates,
others in Cartesian coordinates… Observe:
Polar Equation
Cartesian Equivalent
r cos   2
2
r cos  sin   4
x2
xy  4
r cos   r sin   1
x  y 1
r  1  2r cos 
y  3x  4 x  1  0
r  1  cos 
x  y  2x y
3
2
2
2 x  2 xy  y  0
2
2
2
2
2
2
2
4
2
4
2
2
Relating Polar and Cartesian Coordinates
Find a polar equation for the circle
Support graphically.
Expand and simplify:
x  y  6y  9  9
2
2
x  y  6y  0
Conversion equations:
Algebra:
x   y  3  9
2
2
2
2
r  6r sin   0
2
r  r  6sin    0
r 0
or
r  6sin 
Check the graph!
Relating Polar and Cartesian Coordinates
Find a Cartesian equivalent for the polar equation. Identify
the graph.
(a)
r  4r cos 
2
x  y  4x
2
2
Conversion equations
x  4x  y  0
2
2
x  4x  4  y  4
2
2
 x  2
2
Completing the square
y 4
2
The graph of the equivalent Cartesian equation is a circle
with radius 2 and center (2, 0).
Relating Polar and Cartesian Coordinates
Find a Cartesian equivalent for the polar equation. Identify
the graph.
(b)
4
r
2 cos   sin 
r  2 cos   sin    4
2r cos   r sin   4
2 x  y  4 Conversion equations
y  2x  4
The graph of the equivalent Cartesian equation is a line
with slope 2 and y-intercept –4.
Exploration 2
The polar curves r  a cos n and r  a sin n , where
n is an integer and n  1, are rose curves.
1. Graph r
the curves.
 2 cos n
for
n  2, 4, 6.
Describe
Graph window: [–4.7, 4.7] by [–3.1, 3.1]
The graphs are rose curves with 4 petals when n  2 ,
8 petals when n  4, and 12 petals when n  6 .
2. What is the shortest length a
produce the graphs in (1)?
 -interval can have and still
Shortest interval:
2
Exploration 2
The polar curves r  a cos n and r  a sin n , where
n is an integer and n  1, are rose curves.
3. Based on your observations in (1), describe the graph of
r  2 cos n when n is a nonzero even integer.
The graph is a rose curve with
4. Graph r
the curves.
 2 cos n
for
2 n petals.
n  3, 5, 7.
Describe
Graph window: [–4.7, 4.7] by [–3.1, 3.1]
The graphs are rose curves with 3 petals when n  3 ,
5 petals when n  5, and 7 petals when n  7 .
Exploration 2
The polar curves r  a cos n and r  a sin n , where
n is an integer and n  1, are rose curves.
5. What is the shortest length a
produce the graphs in (4)?
 -interval can have and still
Shortest interval:

6. Based on your observations in (4), describe the graph of
r  2 cos n when n is a nonzero odd integer different
from 1 .
The graph is a rose curve with
n petals.
Guided Practice
Replace the polar equation by an equivalent Cartesian
equation. Then identify or describe the graph.
r  cot  csc 
r sin   cot 
x
y
y
2
y  x A parabola that opens to the right
Guided Practice
Replace the polar equation by an equivalent Cartesian
equation. Then identify or describe the graph.
r  2r cos  sin   1
2
2
r  2  r cos   r sin    1
2
x  y  2 xy  1
2
2
x  2 xy  y  1
2
2
 x  y
1
x  y  1
2
The union of
two lines
Guided Practice
Replace the polar equation by an equivalent Cartesian
equation. Then identify or describe the graph.
r  8sin 
r  8r sin 
2
2
x  y  8y
2
2
x  y  8y  0
2
2
x  y  8 y  16  16
2
2
x   y  4   16 A circle with center (0, 4)
2
and radius 4
Guided Practice
Replace the Cartesian equation by an equivalent polar
equation. Support graphically.
x y 3
r cos   r sin   3
r  cos   sin    3
3
r
cos   sin 
How about the graph?
Guided Practice
Replace the Cartesian equation by an equivalent polar
equation. Support graphically.
x  xy  y  1
2
2
 r cos     r cos   r sin     r sin  
2
2
1
r cos   r cos  sin   r sin   1
2
2
2
2
r  cos   cos  sin   sin    1
2
2
2
r 1  cos  sin    1
2
Graph:
1
r
1  cos  sin 
How about the graph?