9-1: Polar Coordinates
&
9-2: Graphs of Polar Equations
Objectives
•Graph points in polar coordinates.
•Graph simple polar equations.
•Determine the distance between two points with polar
coordinates.
•Graph polar equations.
Real World Application
Before large road construction projects, or even the construction of a
new home, take place, a surveyor maps out characteristics of the
land. A surveyor uses a device called a theodolite to measure
angles. The precise locations of various land features are
determined using distances and the angles measured with the
theodolite.
Polar Graph
Polar Vocabulary
• Point O: pole or origin
• Polar axis: usually the horizontal ray directed right from the pole
• Point P is identified by polar coordinates in the form (r, θ).
|r|: distance from the pole to P
Θ: measure of the angle formed by OP and the polar axis (can
be measured in degrees or radians)
r Values
P(r,θ)
θ
P(r,θ) when r>0
P(r,θ) when r<0
θ
P(r,θ)
Example
Graph each point.
A(-4,0)
B(2,3π/2)
Example
Graph the point.
C(-2,-240°)
Uniqueness
•Polar coordinates are not unique.
•Every point can be represented by infinitely many pairs of
polar coordinates by adding or subtracting 360° or 2π
radians.
Example
How else can you write (2, 120°) using degrees?
(2, -240°)
(2, 480°)
(-2,-60°)
(-2, 300°)
(-2,-420°)
Example
Name four different pairs of polar coordinates that represent point R
on the graph with the restriction that -360°≤r≤ 360°.
(2, 210°)
(2, -150°)
(-2, 30°)
(-2, -330°)
R
Polar Equation
An equation expressed in terms of polar
coordinates is called a polar equation.
Example
Graph the polar equation r=6.
Example
Graph the polar equation θ=5π/6.
Distance Formula in Polar Plane
If
and
polar plane, then
P1 (r1 ,1 )
P2 (r2 ,
are
2)
two points in the
2
2
PP

r

r
1 2
1
2  2r1r2 cos(2  1 ).
r2
r1
Law of Cosines
Surveying Example
If one landmark is 450 feet away and 30° to the left and another
landmark is 600 feet away and 50° to the right. Find the distance
between the two landmarks.
P1 P2  4502  6002  2(450)(600) cos( 50  30)
 684.6 ft
450 ft
600 ft
9-2 Graphs of Polar Equations
Example 1 (p.561)
Example 2a (p. 562)
Example 2b (p.563)
Example 3 (p. 563)
Homework
9-1 p. 558
#6-13 all, #17-39 odds
2nd day:
------------------------------------------p. 558: #14, 41, 43, 45
9-2 p. 565
#11-15 odd, 21