6
CHAPTER 1. COORDINATES AND VECTORS
θ
x
y
P
•
ℓ
O
r
→
Figure 1.5: Polar Coordinates
direction of
ℓ
amounts to a (further) rotation by
π
radians, so the point
with polar coordinates (
r,θ
) also has polar coordinates (
−
r,θ
+
π
).
In fact, while a given geometric point
P
has only one pair of
rectangular
coordinates (
x,y
), it has many pairs of
polar
coordinates. Given (
x,y
),
r
can be either solution (positive or negative) of the equation
r
2
=
x
2
+
y
2
(1.4)
which follows from a standard trigonometric identity. The angle by which
the
x
-axis has been rotated to obtain
ℓ
determines
θ
only up to adding an
even multiple of
π
: we will tend to measure the angle by a value of
θ
between 0 and 2
π
or between
−
π
and
π
, but any appropriate real value is
allowed. Up to this ambiguity, though, we can try to ±nd

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