8.2.3 – Polar Equations and their Graphs

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

• 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

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

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

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

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

• 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

• Example. Graph the polar equation r = 4

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

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”

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

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

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