8.2.3 – Polar Equations and their Graphs

1. 8.2.3 – Polar Equations and their Graphs

2. Polar Equations • Most general definition is an equation in terms of r (radius) and ϴ(measured angle) • Solutions still exist for polar equations, and much like Cartesian equations, we can graph the set of all the solutions

3. So far, we have discussed two parts of the polar system • 1) Converting Cartesian to Polar, vice versa • 2) Graphing Polar points • Just as with Cartesian points, we may need to graph an equation

4. Converting Rectangular to Polar • Note: Rectangular implies Cartesian • Recall from the other day… • x = rcos(ϴ) • y = rsin(ϴ) • To convert rectangular to polar, just use the above substitutions, much like the other day

5. Example. Rewrite the equation x2 – 2x + y2 = 0 in Polar form. • May need to use identities!

6. Example. Convert the rectangular equation x2 + y2 = 12a to polar form.

7. Graphing Polar Equations • Similar to other equations we’ve done before, we may graph polar equations • Some are simple and may be done by hand quickly • Otherwise, we will utilize our graphing calculators to assist us

8. When an equation only contains one variable, r or ϴ, it is simple • 1) If only r, then we can choose any angle we would like • 2) If only ϴ, then we may choose any radius for that value

9. Example. Graph the polar equation r = 4

10. Example. Graph the polar equation ϴ = 2π/3

11. Using Graphing Calculator • Polar equations are often much more complex to graph • Rather than trying to use a table, we will use our calculators to help us • Settings • Mode: • 2nd row should be “RADIAN” • 3rd row should be “POL”

12. Example. Graph the polar equation r = 2sin(ϴ)

13. Example. Graph the polar equation r = 4cos(5ϴ)

14. Assignment • Pg. 629 • 19-29 odd • 47-57 odd (show sketch of graph)