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
x2
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?
0
