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 x 2 – 2x + y 2 = 0 in Polar form. – May need to use identities!

Example. Convert the rectangular equation x 2 + y 2 = 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: – 2 nd row should be “RADIAN” – 3 rd row should be “POL”

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

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

Assignment Pg odd odd (show sketch of graph)