Chapter 12
Polar CoordinatesSo far points have been identified in terms of Cartesian coordinates but there are otherways of specifying points in twodimensions. These other ways involve using a list of twoor three numbers which have a totally different meaning than Cartesian coordinates to spec-ify a point in two or three dimensional space. In general these lists of numbers which havea different meaning than Cartesian coordinates are called curvilinear coordinates. Proba-bly the simplest curvilinear coordinate system is that of polar coordinates. The idea issuggested in the following picture.
x
y
θ
r
(x,y)(r,θ)
You see in this picture, the number r identifies the distance of the point from the origin,(0,0) while θ is the angle shown between the positive x axis and the line from the originto the point. This angle will always be given in radians and is in the interval [0,2π). Thusthe given point, indicated by a small dot in the picture, can be described in terms of theCartesian coordinates (x,y) or the polar coordinates (r,θ). How are the two coordinatessystems related? From the picture,
x = r cos(θ) , y = r sin(θ) . (12.1)
Example 12.0.1 The polar coordinates of a point in the plane are(5, π
6
). Find the Carte-
sian or rectangular coordinates of this point.
From 12.1, x = 5cos(
π
6
)= 5
2
√3 and y = 5sin
(π
6
)= 5
2 . Thus the Cartesian coordinatesare( 5
2
√3, 5
2
).
Example 12.0.2 Suppose the Cartesian coordinates of a point are (3,4). Find the polarcoordinates.
Recall that r is the distance form (0,0) and so r = 5 =√
32 +42. It remains to identifythe angle. Note the point is in the first quadrant. (Both the x and y values are positive.)Therefore, the angle is something between 0 and π/2 and also 3 = 5cos(θ), and 4 =5sin(θ). Therefore, dividing yields tan(θ) = 4/3. At this point, use a calculator or a tableof trigonometric functions to find that at least approximately, θ = .927295 radians.
12.1 Graphs in Polar CoordinatesJust as in the case of rectangular coordinates, it is possible to use relations between thepolar coordinates to specify points in the plane. The process of sketching their graphs isvery similar to that used to sketch graphs of functions in rectangular coordinates. I will onlyconsider the case where the relation between the polar coordinates is of the form, r = f (θ).To graph such a relation, you can make a table of the form
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