78 Chapter 3Transcendental functions
The figure shows that the position of the point P can be specified not only by its
cartesian coordinates (x, y), but also by the pair of numbers (r, θ), the distance rof
the point from the origin and the angle θwhich gives the orientation of the line OP
with respect to the positive x-direction (with the usual convention about sign). The
numbers rand θare called the polar coordinatesof the point in the plane.
4Whereas
the cartesian coordinates can take any real positive and negative values, ris necessarily
positive, with valuesr 1 = 1 0to∞, and the angle θ, has valuesθ 1 = 1 0to2π.
The two sets of coordinates are related bycos 1 θ 1 = 1 x 2 randsin 1 θ 1 = 1 y 2 r, so that
x 1 = 1 r 1 cos 1 θ y 1 = 1 r 1 sin 1 θ (3.27)
and the conversion from polar to cartesian coordinates is straightforward.
EXAMPLE 3.16Find the cartesian coordinates of a point whose polar coordinates
arer 1 = 12 andθ 1 = 1 π 26.
Making use of the values displayed in Table 3.2,
0 Exercises 27, 28
The polar coordinates can be obtained from the pair of equations
However, becausetan 1 θis a periodic function, period π, the angle θis not uniquely
determined by the inverse tangent tan
− 1(y 2 x), and some care needs to be taken in
rxy
y
x
222=+ tanθ=
xr
yr
== =
== =
cos cos
sin sin
θ
θ
2
6
3
2
6
1
π
π
4In his Methodus fluxionum(Method of fluxions), written in about 1671, Newton suggested eight new types of
coordinate system. The polar coordinates were his ‘seventh manner; for spirals’.
P
θ
x
y
r
x
y
O
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Figure 3.17