1. 8.2.1 – Intro to the Polar Coordinate System

2. We already know about the x,y Cartesian plane Now, introduce a new system known as the Polar Coordinate System = points determined by distance from a fixed point AND angle from a fixed direction – Combination of radius, r, and some angle θ

3. The polar system is used extensively in physics, nagivation, and surveying We will use it throughout to help us better determine applications involving angles

4. Conversion The first thing we will do is find a way to convert cartesian to polar, and vice versa To convert from polar to Cartesian, consider the following diagram:

5. What could we write to represent the “x” or horizontal coordinate? What could we write to represent the “y” or vertical coordinate?

6. Conversion Given the polar coodinate (r, θ), x and y are defined as: x = rcosθ y = rsinθ

7. Example. Convert the following polar coordinates to cartesian coordinates: A) (2, -π/3) B) (-3, π/4) C) (-4, 3π)

8. The Other Way To convert the other way around, Cartesian to polar, once again, consider the following diagram

9. Can we come up with an expression for r? Can we find an expression for ϴ?

10. Conversion To convert Cartesian to polar coordinates (r, ϴ); r 2 = x 2 + y 2 tan(ϴ) = y/x

11. Important! When finding the angle θ, we MUST make sure we end in the same quadrant as the cartesian points – Tan is periodic over π – So, may need to add π to our answers

12. Example. Convert the following coordinates from Cartesian to polar. (-3,2)

13. Example. Convert the following Cartesian coordinates to polar coordinates; (√3, -1)

14. Assignment Pg. 629 7-18