1. 1 10.1 Polar Coordinates In this section, we will study the following topics: Converting between polar and rectangular coordinates

2. 2 The Polar Coordinate System The polar axis corresponds to the positive x-axis. The pole corresponds to the origin.

3. 3 Polar Coordinates A point P in the polar coordinate system is represented by an ordered pair. If, then r is the distance of the point from the pole.  is an angle (in degrees or radians) formed by the polar axis and a ray from the pole through the point.

4. 4 Polar Coordinates If, then the point is located units on the ray that extends in the opposite direction of the terminal side of .

5. 5 Example *Plot the point P with polar coordinates

6. 6 Example (r,  )  Polar axis Pole Plot the point with polar coordinates 4

7. 7 Example Plot the point P with polar coordinates P

8. 8

9. 9

10. 10 A)B) C) D)

11. 11 To find the rectangular coordinates for the point, we can use trig functions. Converting from Polar to Rectangular Coordinates

12. 12

13. 13 Give exact answers.

14. 14 Give exact answers.

15. 15 Likewise, we can find the polar coordinates if we are given the rectangular coordinates using the trig functions. Example: Find the polar coordinates (with  in radians to three decimal places) for the point for which the rectangular coordinates are given: (5, 4) Converting from Rectangular to Polar Coordinates (5, 4) x y

16. 16 Converting from Rectangular Coordinates to Polar Coordinates If  is NOT a quadrantal angle: To find , determine the quadrant in which the point lies. If the point is in:

17. 17 Give exact answers, with θ in radians in terms of π.

18. 18 Give exact answers, with θ in radians in terms of π.

19. 19 The TI-84 calculator has handy conversion features built-in. Check out the ANGLE menu. 5: Returns value of r given rectangular coordinates (x, y) 6: Returns value of  given rectangular coordinates (x, y) 7: Returns value of x given polar coordinates (r,  ) 8: Returns value of y given polar coordinates (r,  ) Check the MODE for the appropriate setting for angle measure (degrees vs. radians).

20. 20 End of Section 10.1