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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