Download Polar Coordinate System

Polar Coordinate System
by CHED on June 15, 2017
lesson duration of 2 minutes
under Precalculus
generated on June 15, 2017 at 11:47 am
Tags: Trigonometry
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K-12 Teacher's Resource Community
Generated: Jun 15,2017 07:47 PM
Polar Coordinate System
( 2 mins )
Written By: CHED on July 5, 2016
Subjects: Precalculus
Tags: Trigonometry
Resources
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Content Standard
Key concepts of circular functions, trigonometric identities, inverse trigonometric functions, and the polar coordinate
system
Performance Standard
Formulate and solve accurately situational problems involving circular functions
Apply appropriate trigonometric identities in solving situational problems
Formulate and solve accurately situational problems involving appropriate trigonometric functions
Formulate and solve accurately situational problems involving the polar coordinate system
Learning Competencies
Solve situational problems involving polar coordinate system
Introduction 1 mins
Two-dimensional
coordinate
systems
are
used
describe
a point
a plane.
We
previously
used
Cartesian
Two-dimensional
coordinate
systems
are
used
to to
describe
a point
in in
a plane.
We
previously
used
thethe
Cartesian
or or
rectangularcoordinate
coordinatesystem
systemtotolocate
locatea apoint
pointin inthe
theplane.
plane.That
Thatpoint
pointis isdenoted
denotedbyby( (x
x, y), where x is
is the
the signed
signed
rectangular
distance of the point from the y-axis, and y is the signed distance of the point from the x-axis. We sketched the graphs
of equations
(lines,
circles,
parabolas,
ellipses,
hyperbolas)
functions
(polynomial,
rational,
exponential,
of equations
(lines,
circles,
parabolas,
ellipses,
andand
hyperbolas)
andand
functions
(polynomial,
rational,
exponential,
logarithmic,
logarithmic, trigonometric,
trigonometric, and
and inverse
inverse trigonometric)
trigonometric) in
in the
the Cartesian
Cartesian coordinate
coordinate plane.
plane. However,
However, itit is
is often
often convenient
convenient
to locate a point based on its distance from a fixed point and its angle with respect to a fixed ray. Not all equations can
begraphed
graphedeasily
easilyusing
usingCartesian
Cartesiancoordinates.
coordinates.InInthis
thislesson,
lesson,we
wealso
alsouse
useanother
anothercoordinate
coordinatesystem,
system,which
whichcan
canbe
be
be
presented in dartboard-like plane as shown below.
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Polar Coordinated of a Point
We now introduce the polar coordinate system.
system. It is composed of a fixed point called the pole (which is the origin in the
Cartesian coordinate system) and a fixed ray called the polar axis (which is the nonnegative x-axis).
In the polar coordinate system, a point is described by the ordered pair (r, ?), where the radial coordinater
coordinater refers to the
directed distance of the point from the pole and the angular coordinate ? refers to a directed angle (usually in radians)
from the polar axis to the segment joining the point and the pole.
Because a point in polar coordinate system is described by an order pair of radial coordinate and angular coordinate, it
willbebemore
moreconvenient
convenienttotogeometrically
geometricallypresent
presentthe
thesystem
systeminina a polar plane
plane, ,which
whichserves
servesjust
justlike
likethe
theCartesian
Cartesian
will
plane.In
Inthe
thepolar
polarplane
planeshown
shownbelow,
below,instead
insteadof
ofrectangular
rectangulargrids
gridsininthe
theCartesian
Cartesianplane,
plane,we
wehave
haveconcentric
concentriccircles
circles
plane.
common
center
at the
to identify
easily
distance
(radial
coordinate)
angular
withwith
common
center
at the
polepole
to identify
easily
the the
distance
fromfrom
the the
polepole
(radial
coordinate)
andand
angular
raysrays
emanating from the pole to show the angles from the polar axis (angular coordinate).
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Example 3.9.1.
3.9.1. Plot
Plot the
the following
following points
points in
in one
one polar
polar plane:
plane: A(3, pi/3), B(1, 5pi/6), C(2, 7pi/6), D(4, 19pi/12), E(3, ?pi),
Example
F(4, ?7pi/6), G(2.5, 17pi/4 ), H(4, 17pi/6), and I(3,?5pi/3).
Solution.
As
As seen
seen in
in the
the last
last example,
example, unlike
unlike in
in Cartesian
Cartesian plane
plane where
where aa point
point has
has aa unique
unique Cartesian
Cartesian coordinate
coordinate representation,
representation,
point in
in polar
polar plane
plane have
have infinitely
infinitely many
many polar
polar coordinate
coordinate representations.
representations. For
For example,
example, the
the coordinates
coordinates (3,
(3, 4)
4) in
in the
the
aa point
Cartesian
to exactly
in plane,
the plane,
this particular
hasrectangular
no rectangular
coordinate
Cartesian
planeplane
referrefer
to exactly
one one
pointpoint
in the
and and
this particular
pointpoint
has no
coordinate
representationsother
otherthan
than(3,
(3,4).
4).However,
However,the
thecoordinates
coordinates(3,
(3,pi/3)
pi/3)ininthe
thepolar
polarplane
planealso
alsorefer
refertotoexactly
exactlyone
onepoint,
point,
representations
butthis
thispoint
pointhas
hasother
otherpolar
polarcoordinate
coordinaterepresentations.
representations.For
Forexample,
example,the
thepolar
polarcoordinates
coordinates(3,?5pi/3),
(3,?5pi/3),(3,
(3,7pi/3),
7pi/3),(3,
(3,
but
13pi/3), and (3, 19pi/3) all refer to the same point as that of (3, pi/3).
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In
In polar
polar coordinate
coordinate system,
system, itit is
is possible
possible for
for the
the coordinates
coordinates (r,
(r, ?)
?) to
to have
have aa negative
negative value
value of
of r. In this case, the point is
|r| units from the pole in the opposite direction of the terminal side of ?, as shown in Figure 3.36.
Example 3.9.2. Plot the following points in one polar plane: A(?3, 4pi/3), B(?4, 11pi/6), C(?2,?pi), and D(?3.5,?7pi/4).
Solution. As described above, a polar point with negative radial coordinate lies on the opposite ray of the terminal side
of ?.
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Seatwork/Homework 3.9.1
1. Plot the following points in one polar plane:
Answer:
2. Give the polar coordinates (r, ?) with indicated properties that represent the same point as the given polar
coordinates
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From Polar to Rectangular, and Vice Versa
Wenow
nowhave
havetwo
twoways
waystotodescribe
describepoints
pointsonona aplane
plane– –whether
whethertotouse
usethe
theCartesian
Cartesiancoordinates
coordinates( (x
x, y) or the polar
We
coordinates (r, ?). We now derive the conversion from one of these coordinate systems to the other.
We superimpose the Cartesian and polar planes, as shown in the following diagram.
is represented
represented by
by the
the polar
polar coordinates
coordinates ((rr,, ?).
?). From
From Lesson
Lesson 3.2
3.2 (in
(in particular,
particular, the
the boxed
boxed definition
definition
Suppose a point P is
on page 138), we know that
x = r cos ? and y = r sin ?.
Example 3.9.3. Convert the polar coordinates (5, pi) and (?3, pi/6) to Cartesian coordinates.
Solution.
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Teaching Notes
One can
can also
also easily
easily convert
convert the
the polar
polar coordinates
coordinates (5,pi)
(5,pi) to
to its
its corresponding
corresponding rectangular
rectangular coordinates
coordinates (?5,
(?5, 0)
0) by
by simply
simply
One
plotting the point.
As explained on page 254 (right after Example 3.9.1), we expect that there are infinitely many polar coordinate
representations that correspond to just one given rectangular coordinate representation. Although we can actually
determine all of them, we only need to know one of them and we can choose r ) 0.
Suppose a point P is represented by the rectangular coordinates (x, y). Referring back to Figure 3.37, the equation of
the circle is
x^2 + y^2 = r^2 => r = square root of (x^2 + y^2).
and the
the point
point is
is the
the pole.
pole. The
The pole
pole has
has coordinates
coordinates (0,
(0, ?),
?), where
where ?? is
is any
any
We now determine ?. If x = y = 0, then r == 00 and
real number.
If x == 00 and
and yy =/=
=/= 0,
0, then
then we
we may
may choose
choose ?? to
to be
be either
either pi/2
pi/2 or
or 3pi/2
3pi/2 (or
(or their
their equivalents)
equivalents) depending
depending on
on whether
whether yy >> 00
or y < 0, respectively.
Now, suppose x =/= 0. From the boxed definition again on page 138, we know that
tan ? = y/x
where ? is an angle in standard position whose terminal side passes through the point (x
(x, y).
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Example 3.9.4. Convert each Cartesian coordinates to polar coordinates (r, ?), where r > 0.
(1) (?4, 0)
(2) (4, 4)
(3) (?3,?square root of 3)
(4) (6,?2)
(5) (?3, 6)
(6) (?12,?8)
Solution. (1) (?4, 0) ?> (4, pi)
Teaching Notes
Plotting the points on the superimposed Cartesian and polar planes is a quicker approach in converting rectangular
coordinates to polar.
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Teaching Notes
Recall that tan^?1(?2) is in QIV.
Teaching Notes
We may also use ? = tan^?1 2/3 ? pi.
Seatwork/Homework
1. Convert each polar coordinates to Cartesian coordinates.
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2. Convert each Cartesian coordinates to polar coordinates (r, ?), where r > 0.
From the preceding session, we learned how to convert polar coordinates of a
point to rectangular and vice versa using the following conversion formulas:
r2 = x2 + y2, tan 0 = y/x, x = r cos 0, and y = r sin 0.
Because a graph is composed of points, we can identify the graphs of some equations in terms of r and 0.
As a quick illustration, the polar graph of the equation r = 2 ? 2sin 0 consists of all points (r
(r, 0) that satisfy the equation.
Some of these points are (2, 0), (1, pi/6), (0, pi/2), (2,pi), and (4, 3pi/ 2).
Example 3.9.5. Identify the polar graph of r = 2, and sketch its graph in the polar plane.
Solution. Squaring the equation, we get r2 = 4. Because r2 = x2 + y2, we have x2 + y2 == 4,
4, which
which is
is aa circle
circle of
of radius
radius 22
andwith
withcenter
centeratatthe
theorigin.
origin.Therefore,
Therefore,the
thegraph
graphofof r = =2 2is isa acircle
circleofofradius
radius2 2with
withcenter
centeratatthe
thepole,
pole,asasshown
shown
and
below.
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In the previous example, instead of using the conversion formula r2 = x2+y2, we may also identify the graph of r = 2 by
observingthat
thatits
itsgraph
graphconsists
consistsofofpoints
points(2,
(2, 0)) for
for all
all 0. .InInother
otherwords,
words,the
thegraph
graphconsists
consistsofofall
allpoints
pointswith
withradial
radial
observing
distance 2 from the pole as 0 rotates around the polar plane. Therefore, the graph of r = 2 is indeed a circle of radius 2
as shown.
Example 3.9.6. Identify and sketch the polar graph of 0 = ?5pi/4.
Solution. The graph of 0 = ?5pi/4 consists of all points (r
(r, ?5pi/4) for rER. If r > 0, then points (r
(r, ?5pi/4) determine a ray
from the pole with angle ?5pi/4 from the polar axis. If r = 0, then (0,?5pi/4) is the pole. If r < 0, then the points (r, ?5pi/4)
determine a ray in opposite direction to that of r > 0. Therefore, the graph of ? = ?5?4 is a line passing through the pole
and with angle ?5?4 with respect to the polar axis, as shown below.
Example 3.9.7. Identify (and describe) the graph of the equation r = 4sin 0.
Solution.
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r = 4sin 0
r2 = 4r
4r sin 0
x2 + y2 = 4y
x2 + y2 ? 4y = 0
x2 + (y ? 2)2 = 4
Therefore, the graph of r = 4sin 0 is a circle of radius 2 and with center at (2, pi/2).
Example 3.9.8. Sketch the graph of r = 2? 2 sin 0.
Solution. We construct a table of values.
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This heart-shaped curve is called a cardioid.
Example 3.9.9. The sound-pickup capability of a certain brand of microphone is described by the polar equation r = ?4
cos 0, where |r
|r| gives the sensitivity of the microphone to a sound coming from an angle 0 (in radians).
(1) Identify and sketch the graph of the polar equation.
(2) Sound coming from what angle 0E [0, pi] is the microphone most sensitive to? Least sensitive?
Solution. (1) r = ?4 cos 0
r2 = ?4 cos 0
x2 + y2 = ?4x
?4x
x2 + 4x
4x + y2 = 0
(x + 2)2 + y2 = 4
This is a circle of radius 2 and with center at (2, pi).
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(2) We construct a table of values.
From the table, the microphone is most sensitive to sounds coming from angles ? = 0 and 0 = pi, and least sensitive to
sound coming from an angle 0 = pi/2.
Seatwork/Homework 3.9.3
1. Identify (and describe) the graph of each polar equation.
(a) 0 = 2pi/3
Answer: Line passing through the pole with angle 2pi/3 with respect to the polar axis
(b) r = ?3
Answer: Circle with center at the pole and of radius 3
(c) r = 2sin 0
Answer: Circle of radius 1 and with center at (1, pi/2)
(d) r = 3cos ?
Answer: Circle of radius 1.5 and with center at (1.5, 0)
(e) r = 2+2cos 0
Answer: A cardioid
2. Sketch the graph of each polar equation.
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(a) r = ?3
(b) r = ?2 sin 0
(c) r = 2 + 2sin 0
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(d) r = 4cos 0
3. The sound-pickup capability of a certain brand of microphone is described by the polar equation
r = 1.5(1 + cos ?),
where |r
|r| gives the sensitivity of the microphone of a sound coming from an angle 0 (in radians).
(a) Identify and sketch the graph of the polar equation.
Answer: A cardioid
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(b) Sound coming from what angle 0E [0, 2pi) is the microphone most sensitive to? Least sensitive?
Answer: Most sensitive at 0 = 0; least sensitive at 0 = pi
Exercises 1 mins
1. Plot the following points in one polar plane:
Answer:
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2. Give the polar coordinates (r,
given polar coordinates.
) with indicated properties that represent the same point as the
3. Convert each polar coordinates to Cartesian coordinates.
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4. Convert each Cartesian coordinates to polar coordinates (r,
2
), where r ) 0 and 0 ?
.
5. Identify and sketch the graph of each polar equation.
(a)
=–
/3
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Answer: A line passing through the pole and with angle –
/3 with respect to the polar axis
(b) r = –3 sin
Answer: A circle tangent to the x-axis with center at (0,?1.5)
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(c) r = cos
Answer: A circle tangent to the y-axis with center at (0.5, 0)
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(d) r = 2– 2 cos
Answer: A cardioid
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(e) r = 1+sin
Answer: A cardioid
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6. The graph of the polar equation r = 2cos2
is a four-petaled rose. Sketch its graph.
Answer:
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*7. A comet travels on an elliptical orbit that can be described by the polar
equation
r = 1.164 / 1 + 0.967 sin
with respect to the sun at the pole. Find the closest distance between the sun and the comet
Answer: Closest distance occurs when sin
= 1, so r = 1.164 / 1.967 ? 0.59 units
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*8. Polar equations are also used by scientists and engineers to model motion of satellites orbiting the Earth. One
satellite follows the path
r = 36210 / 6 – cos
,
where r is the distance in kilometers between the center of the Earth and the satellite, and
angular measurement in radians with respect to a fixed predetermined axis
(a) At what value of
closest distance?
2 [0, 2
is the
) is the satellite closest to Earth, and what is the
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Answer: The satellite is closest to Earth when cos
=
= –1, and this occurs when
. The closest distance is, therefore, r = 36210 6–(–1)
5182.86 kilometers.
(b) How far away from Earth can the satellite reach?
Answer: The satellite can reach as far as r = 36210 6–1
7242 km away from the Earth
9. The graph of the polar equation
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