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Unformatted text preview: 7. The ﬁres are about 17456 feet apart. (Try to avoid rounding errors.) 782 11.4 Applications of Trigonometry Polar Coordinates In Section 1.1, we introduced the Cartesian coordinates of a point in the plane as a means of
assigning ordered pairs of numbers to points in the plane. We deﬁned the Cartesian coordinate
plane using two number lines – one horizontal and one vertical – which intersect at right angles at a
point we called the ‘origin’. To plot a point, say P (−4, 2), we start at the origin, travel horizontally
to the left 4 units, then up 2 units. Alternatively, we could start at the origin, travel up 2 units,
then to the left 4 units and arrive at the same location. For the most part, the ‘motions’ of the
Cartesian system (over and up) describe a rectangle, and most points be thought of as the corner
diagonally across the rectangle from the origin.1 For this reason, the Cartesian coordinates of a
point are often called ‘rectangular’ coordinates.
3 P (−4, 2) 1
−4 −3 −2 −1
−1 1 2 3 4 x −2
−4 In this section, we introduce a new system for assigning coordinates to points in the plane – polar
coordinates. We start with a point, called the pole, and a ray called the polar axis.
Pole Polar Axis Polar coordinates consist of a pair of numbers, (r, θ), where r represents a directed distance from
the pole2 and θ is a measure of rotation from the polar axis. If we wished to plot the point P with
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