1. Polar Coordinates Lesson 10.5

2. Points on a Plane Rectangular coordinate system  Represent a point by two distances from the origin  Horizontal dist, Vertical dist Also possible to represent different ways Consider using dist from origin, angle formed with positive x-axis r θ (x, y) (r, θ)

3. Plot Given Polar Coordinates Locate the following

4. Find Polar Coordinates What are the coordinates for the given points? B A C D A = B = C = D =

5. Converting Polar to Rectangular Given polar coordinates (r, θ)  Change to rectangular By trigonometry  x = r cos θ y = r sin θ Try = ( ___, ___ ) θ r x y

6. Converting Rectangular to Polar Given a point (x, y)  Convert to (r, θ) By Pythagorean theorem r 2 = x 2 + y 2 By trigonometry Try this one … for (2, 1)  r = ______  θ = ______ θ r x y

7. Polar Equations States a relationship between all the points (r, θ) that satisfy the equation Exampler = 4 sin θ  Resulting values θ in degrees Note: for (r, θ) It is θ (the 2 nd element that is the independent variable Note: for (r, θ) It is θ (the 2 nd element that is the independent variable

8. Graphing Polar Equations Set Mode on TI calculator  Mode, then Graph => Polar Note difference of Y= screen

9. Graphing Polar Equations Also best to keep angles in radians Enter function in Y= screen

10. Graphing Polar Equations Set Zoom to Standard,  then Square

11. Try These! For r = A cos B θ  Try to determine what affect A and B have r = 3 sin 2θ r = 4 cos 3θ r = 2 + 5 sin 4θ

12. 12 Finding dy/dx We know  r = f(θ) and y = r sin θ and x = r cos θ Then And

13. 13 Finding dy/dx Since Then

14. 14 Example Given r = cos 3θ  Find the slope of the line tangent at (1/2, π/9)  dy/dx = ?  Evaluate

15. Define for Calculator It is possible to define this derivative as a function on your calculator 15

16. 16 Try This! Find where the tangent line is horizontal for r = 2 cos θ Find dy/dx Set equal to 0, solve for θ

17. Assignment Lesson 10.4 Page 736 Exercises 1 – 19 odd, 23 – 26 all Exercises 69 – 91 EOO