Ch. 8.2 : Polar coordinates and Polar Equations

In this section, we will

1. define the polar coordinate system

2. look at coordinate conversion (i.e. from Cartesian ⇔ Polar )

3. look at various forms of polar equations

4. graph certain polar equations.

A point P in the polar coordinate system.

(r , θ) are polar coordinates of a point P, where r is the length ofOP, and θ is the angle, measured counterclockwise.

Example 1Plot the point given by the following polar coordinates.

1. (2, 3π4 )2. (3,−5π

2 )3. (−1, 4π3 )

Coordinate Conversion

1. To convert from polar to Cartesian coordinates: givenP = (r , θ),

x = r cos θ, y = r sin θ.

2. To convert from Cartesian to polar: given P = (x , y)

r2 = x2 + y2, tan θ =y

x(x 6= 0)

Example 2 : Coordinate ConversionConvert the following points from polar to Cartesian coordinates.

1. (2,−π3 )

2. (−3,−π4 )

Example 3Convert the following points from Cartesian to polar coordinates.

1. (−3, 2)2. (√

3,−1)

Example 4 : The form of polar equations

Rewrite the equation x2 − 2x + y2 = 0 in polar form.

Example 5

Rewrite the equation 2r = sec θ in rectangular coordinates.

Example 6 : Graphing Polar equationsSketch the graphs of the following polar equations, and thenconvert the equations to rectangular coordinates.

1. r = 32. θ = 2π

3

Example 7Sketch the graph of the following polar equation, and convert theequation to rectangular coordinates:

r = 2 cos θ.

ExampleSketch the graph of the following polar equation, and convert theequation to rectangular coordinates:

r = 8 sin θ.

HW for Ch 8.2

Show all work to get credit2, 5, 7, 9, 13-15, 19, 21, 22, 27, 28, 31, 34, 36, 42, 45, 46