Distance Between Two Polar
Coordinates
Andrea Hayes
Art Fortgang
Bradley Hughes
Brenda Meery
Larry Ottman
Lori Jordan
Mara Landers
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Printed: January 22, 2013
AUTHORS
Andrea Hayes
Art Fortgang
Bradley Hughes
Brenda Meery
Larry Ottman
Lori Jordan
Mara Landers
www.ck12.org
C ONCEPT
Concept 1. Distance Between Two Polar Coordinates
1
Distance Between Two
Polar Coordinates
Here you’ll learn how to find the distance between two points that are plotted in a polar coordinate system.
When playing a game of darts with your friend, the darts you throw land in a pattern like this
You and your friend decide to find out how far it is between the two darts you threw. If you know the positions of
each of the darts in polar coordinates, can you somehow find a formula to let you determine the distance between
the two darts?
At the end of this Concept, you’ll be able to answer this question.
Watch This
MEDIA
Click image to the left for more content.
1
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DistanceFormulain Polar Plane
Guidance
Just like the Distance Formula for x and y coordinates, there is a way to find the distance between two polar
2
2
2
coordinates.
p One way that we know how to find distance, or length, is the Law of Cosines, a = b + c − 2bc cos A
or a = b2 + c2 − 2bc cos A. If we have two points (r1 , θ1 ) and (r2 , θ2 ), we can easily substitute r1 for b and r2 for
c. Asqfor A, it needs to be the angle between the two radii, or (θ2 − θ1 ). Finally, a is now distance and you have
d=
r12 + r22 − 2r1 r2 cos(θ2 − θ1 ).
Example A
Find the distance between (3, 60◦ ) and (5, 145◦ ).
Solution:
q After graphing these two points, we have a triangle. Using the new Polar Distance Formula, we have
d = 32 + 52 − 2(3)(5) cos 85◦ ≈ 5.6.
Example B
Find the distance between (9, −45◦ ) and (−4, 70◦ ).
Solution: This one is a little trickier than the last example because we have negatives. The first point would be
plotted in the fourth quadrant and is equivalent to (9, 315◦ ). The second point would be (4, 70◦ ) reflected across the
pole, or (4, 250◦ ). Use these two values of θ for the
qformula. Also, the radii should always be positive when put into
the formula. That being said, the distance is d =
92 + 42 − 2(9)(4) cos(315 − 250)◦ ≈ 8.16.
Example C
Find the distance between (2, 10◦ ) and (7, 10◦ ).
Solution: This problem is straightforward from looking at the relationship between the points. The two points lie
at the same angle, so the straight line distance between them is 7 − 2 = 5. However, we can confirm this using the
distance formula:
q
√
√
d = 22 + 72 − 2(2)(7) cos 0◦ = 4 + 49 − 28 = 25 = 5.
Vocabulary
Polar Coordinates: A set of polar coordinates are a set of coordinates plotted on a system that uses the distance
from the origin and angle from an axis to describe location.
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Concept 1. Distance Between Two Polar Coordinates
Guided Practice
1. Given P1 and P2 , calculate the distance between the points.
P1 (1, 30◦ ) and P2 (6, 135◦ )
2. Given P1 and P2 , calculate the distance between the points.
P1 (2, −65◦ ) and P2 (9, 85◦ )
3. Given P1 and P2 , calculate the distance between the points.
P1 (−3, 142◦ ) and P2 (10, −88◦ )
Solutions:
1. Use P1 P2 =
q
r12 + r22 − 2r1 r2 cos(θ2 − θ1 ).
P1 P2 =
q
12 + 62 − 2(1)(6) cos(135◦ − 30◦ )
P1 P2 ≈ 6.33 units
2. Use P1 P2 =
q
r12 + r22 − 2r1 r2 cos(θ2 − θ1 ).
P1 P2 =
q
22 + 92 − 2(2)(9) cos 150◦
= 10.78
3. Use P1 P2 =
q
r12 + r22 − 2r1 r2 cos(θ2 − θ1 ).
P1 P2 =
q
32 + 102 − 2(3)(10) cos(322 − 272)◦
= 8.39
Concept Problem Solution
Using the Distance Formula for points in a polar plot, it is possible to determine the distance between the 2 darts:
d=
q
32 + 62 − 2(4)(6) cos 45◦
p
d = 9 + 36 − 48(.707)
√
d = 45 − 33.936
d ≈ 11.064
Practice
Find the distance between each set of points.
1. (1, 150◦ ) and (2, 130◦ )
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2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
4
(4, 90◦ ) and (5, 210◦ )
(6, 60◦ )and (2, 90◦ )
(2, 120◦ ) and (1, 150◦ )
(7, 210◦ ) and (4, 300◦ )
(−4, 120◦ ) and (2, 100◦ )
(3, −90◦ ) and (5, 150◦ )
(−4, −30◦ ) and (3, 250◦ )
(7, −150◦ ) and (4, 130◦ )
(−2, 300◦ ) and (2, 10◦ )
Find the length of the arc between the points (3, 40◦ ) and (3, 150◦ ).
Find the length of the arc between the points (1, 10◦ ) and (1, 70◦ ).
Find the area of the sector created by the origin and the points (4, 20◦ ) and (4, 110◦ ).
Find the area of the sector created by the origin and the points (2, 100◦ ) and (2, 180◦ ).
Find the area of the sector created by the origin and the points (5, 110◦ ) and (5, 160◦ ).