Polar Coordinates
Section 9-1
Polar Coordinates
Definition
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„
„
„
The position of an object using the distance from a fixed point and
an angle made with a fixed ray from that point uses a polar
coordinate system.
In a polar coordinate system, a fixed point O is called the pole or
origin.
The polar axis is usually a horizontal ray directed toward the right
from the pole.
The location of a point P in the polar coordinate system can be
identified by polar coordinates in the form (r, θ ).
The polar graph
Cartesian Plane and the Polar
Plane
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Graph the point P (1, ½) on a coordinate plane. What is the length of
Op? What is the angle that OP makes with the x axis?
O
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.P
How does this relate to the polar coordinates?
Positive and Negative Values of r
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If r>0, then θ is the measure of any angle in standard position that
has OP as its terminal side.
. P (r, θ)
θ
O
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If r<0, then θ is the measure of any angle that has the ray opposite
OP as its terminal side.
X
. P (r,θ)
Examples
Every point can be represented many different
ways
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The polar coordinates of a point are not unique. This happens
because any angle in standard position is coterminal with infinitely
many other angles. You can use multiples of 360 degrees to create
another ordered pair. You can also use the opposite r value and an
angle changed by 180 degrees.
0
Below are six examples of how you can write the point (2, 120 ).
There are infinitely more examples for this point.
Example
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Name four different pairs of polar coordinates that represent point S
on the graph with the restriction that -360 ≤ θ≤360. Make sure to
choose values that stay within this restriction.
0
0
0
0
(2, -150 ), (2, 210 ), (-2, 30 ), (-2, -330 )
Polar Equation
An equation expressed in terms of polar
coordinates is called a polar equation.
„ r=k and θ=k result in simple graphs just
like x=k and y=k do in rectangular
coordinates (vertical and horizontal lines).
„ In polar coordinates the graph r=k is a
circle and the graph θ=k is a line through
the origin.
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Example
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Graph r = -3
π
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Graph θ=
5π
6
Distance formula in the polar plane
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Example
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If two landmarks are 700 feet away and 40 degrees to the left (positive
angle), and 350 feet away and 35 degrees to the right (negative angle),
what is the distance between the landmarks?
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(700, 40) (350, -35)
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LM = use the distance formula in
polar plane
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≈ 697 feet
. M (700,40)
. L (350, 35)
HW # 8
: Section 9-1
„ Pp. 558-560
„ #16-23 all, 25, 28, 29,32,33,41
„ 2929,32,33,41
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