1. Section 10.4 – Polar Coordinates and Polar Graphs

2. Introduction to Polar Curves Parametric equations allowed us a new way to define relations: with two equations. Parametric curves opened up a new world of curves: Polar coordinates will introduce a new coordinate system.

3. Introduction to Polar Curves You have only been graphing with standard Cartesian coordinates, which are named for the French philosopher-mathematician, Rene Descartes.

4. Polar Coordinates

5. Example 1

6. Example 2

7. Example 3 r 0

8. The Relationships Between Polar and Cartesian Coordinates Right triangles are always a convenient shape to draw. Using Pythagorean Theorem…

9. The Relationships Between Polar and Cartesian Coordinates You can use a reference angle to find a relationship but that would require an extra step. Instead, compare the coordinates to the unit circle coordinates.

10. The Relationships Between Polar and Cartesian Coordinates Therefore: (Remember tangent is also the slope of the radius.)

11. Conversion Between Polar and Cartesian Coordinates When converting between coordinate systems the following relationships are helpful to remember: NOTE: Because of conterminal angles and negative values of r, there are infinite ways to represent a Cartesian Coordinate in Polar Coordinates.

12. Example 1

13. Example 2 NOTE: In Cartesian coordinates, every point in the plane has exactly one ordered pair that describes it.

14. Example 3 A circle centered at (0,2) with a radius of 2 units.

15. Conversion Between Polar Equations and Parametric Equations Since we can easily convert a polar equation into parametric equations, the calculus for a polar equation can be performed with the parametrically defined functions. The slope of tangent lines is dy/dx not dr/dΘ.

16. Example Parametric Equations: Find dy/dx not dr/dΘ: Find the point: Find the equation:

17. Example (Continued) Parametric Equations: Find dy/dt and dx/dt: Use the Arc Length Formula:

18. Example (Continued) Parametric Equations: Find dy/dx: Find d 2 y/dx 2 : Since the second derivative is positive, the graph is concave up.

19. Alternate Formula for the Slope of a Tangent Line of a Polar Curve If you do not want to easily convert a polar equation into parametric equations, you can always memorize another formula...

20. Alternate Arc Length Formula for Polar Curves If you do not want to easily convert a polar equation into parametric equations, you can always memorize another formula...