1. 9-1: Polar Coordinates & 9-2: Graphs of Polar Equations

2. Objectives Graph points in polar coordinates.
Graph simple polar equations.
Determine the distance between two points with polar coordinates.
Graph polar equations.

3. 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.

4. Polar Graph

5. 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)

6. r Values
P(r,θ)
P(r,θ) when r>0
P(r,θ) when r<0 θ θ
P(r,θ)

7. Example
Graph each point.
A(-4,0)
B(2,3π/2)

8. Example
Graph the point.
C(-2,-240°)

9. 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.

10. Example How else can you write (2, 120°) using degrees? (2, -240°)
(2, 480°)
(-2,-60°)
(-2, 300°)
(-2,-420°)

11. 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

12. Polar Equation
An equation expressed in terms of polar coordinates is called a polar equation.

13. Example
Graph the polar equation r=6.

14. Example
Graph the polar equation θ=5π/6.

15. Distance Formula in Polar Plane
If and are two points in the polar plane, then
Law of Cosines r2 r1

16. 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.
450 ft
600 ft

17. 9-2 Graphs of Polar Equations

18. Example 1 (p.561)

19. Example 2a (p. 562)

20. Example 2b (p.563)

21. Example 3 (p. 563)

22. -------------------------------------------
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