Pangasinan State University
Bayambang Campus
Bayambang, Pangasinan
S.Y. 2021 – 2022
A Detailed Lesson Plan
in
Mathematics
(Graphs of Equations in Polar Coordinates)
Prepared by:
Rafael S. Macaranas
Student Teacher
Prepared to:
Mrs. Irma Mirasol C. Ferrer
Supervising Instructor
Date of Passing:
May 28, 2022
Date of Teaching:
June 27, 2022
I.
Objectives
At the end of a 60 – minute lesson, at least 75% of the students should be
able to achieve the following with at least 75% level of proficiency:
a. define what is polar equation;
b. recognize the equations of different classic curves by the position and
value of the numbers, cosine and sine; and
c. determine how the graph of a polar equation will look by the value of
these terms;
II.
Subject Matter
A. Topic
:
Graphs of Equations in Polar Coordinates
B. Reference
:
Aufmann, R. N., & Nation, R. D. (2014). Algebra and
Trigonometry
(8th
Edition).
Cengage
Learning
https://psu.vitalsource.com/books/9781285965833. pp. 692 – 699.
C. Materials
:
D. Values Integration:
III.
PowerPoint presentation
Perseverance
Procedures
Teacher’s Activity
A.
Preliminary Activities
Greetings
Opening Prayer
Checking of Attendance
Collecting of Assignments
Students’ Activity
US.
B.
Review
I will show 2 order pairs (r,
𝛳) to recall their previous
knowledge
about
polar
coordinates.
I will ask for 2 volunteer
students to plot the order pair
in a polar coordinate.
Direction:
Plot the order pair (r, 𝛳) in
polar coordinates.
Answer:
𝜋
1. (4, )
3
𝜋
2. (3√2, )
4
Answer:
C.
Motivation
FIRST LETTERS GAME
Students
are
shown
different pictures and they
need to take the first letter
from each picture to form the
mystery word.
1. 6 letters
Ans. Circle
2. 7 letters
Ans. Limacon
3. 2 words and 10 letters
Ans. Rose Curves
4. 11 letters
Ans. Lemniscates
Students who will get answer
will receive an award.
D.
Presentation of the Lesson
There is a picture posted
on each question. Students
will give feedbacks on what
they observe.
From the pictures shown, The topic for today is all about graphs
what do you think is our topic and different curves form by an
for today?
E.
equation.
Development of the Lesson
A polar equation is an
equation in r and 𝛳. A solution
to a polar equation is an
ordered
pair
(r,
𝛳)
that
satisfies the equation. The
graph of a polar equation is
the set of all points whose
ordered pairs are solutions of
the equation.
In graphing polar equations
in polar coordinates, there will
be different curves can be
drawn. Just like in rectangular
coordinate system, where you
can plot curves by assigning
different
points
and
connecting them in the given
equation. Table of values can
be
used
equations.
to
graph
polar
Polar Equation of a Line
Graph:
The graph of the polar
𝜋
equation 𝛳 =
6
𝜋
6
𝜋
6
𝜋
6
𝜋
6
𝜋
6
𝜋
6
𝜋
6
𝜋
6
𝜋
6
𝜋
6
r
-4
-3
-2
-1
0
1
2
3
4
is a line.
Because 𝛳 is equated to
constant, 𝛳 is
𝛳
radian from
the polar axis for all values of
r. The graph is a line that
makes an angle of
𝜋
6
radian
(30°) from the polar axis.
In general, the graph of
𝛳
= α is a line through the pole
at an angle of α from the polar
axis.
Polar Equations of a Circle
The graph of the equation
Graph:
𝛳
0
𝜋
6
𝜋
3
𝜋
2
2𝜋
3
5𝜋
6
𝜋
5𝜋
4
3𝜋
2
7𝜋
4
2𝜋
r
4
4
4
4
4
4
4
4
4
4
4
r = α is a circle with center at
the pole and radius α.
The graph of the polar
equation r = 4 is circle.
Because
r
is
equated
to
constant, 4 will be the radius
with
center
at
the
pole.
Moreover, r is independent so
whatever the value of 𝛳, r will
always equal to 4
The graph of the equation r
= α cos 𝛳 is a circle that is
Graph:
symmetric with respect to the
line 𝛳 = 0. If α = +, the circle
𝛳
0
𝜋
6
𝜋
3
r
2
√3
1
is directed to the right. If α = -,
𝜋
2
0
5𝜋
4
3𝜋
2
7𝜋
4
−√2
0
√2
2𝜋
3
5𝜋
6
-1
−√3
5𝜋
6
𝜋
5𝜋
4
1
0
−1.41
𝜋
2
the circle is directed to the
2𝜋
2
left.
The graph of the polar
equation r = 2cos𝛳 is circle
directed to the right that is
symmetric with respect to the
line 𝛳 = 0.
The graph of r = α sin 𝛳 is
Graph:
a circle that is symmetric with
respect to the line
𝜋
𝛳 = 2 . If
α = +, the circle is directed
upward. If α = -, the circle is
directed downward.
The graph of the polar
equation r = 2sin𝛳 is circle
directed
upward
that
is
symmetric with respect to the
𝜋
line 𝛳 = 2 .
𝛳
0
𝜋
6
r
0
1
𝜋
3
𝜋
2
2𝜋
3
1.7
2
1.7
3
3
3𝜋
2
2
7𝜋
4
−1.41
2𝜋
0
Just
as
there
are
specifically named curves in
an
xy-coordinate
(such
ellipse),
as
system
parabola
there
are
and
named
curves in an r𝛳-coordinate
system. Three of the many
types are the limaçon, the
rose
curve,
and
the
lemniscate
Polar equations of a Limaçon
The graph of the equation
r = a + b cos𝛳
r = a + b cos𝛳 is a limaçon
that is symmetric with respect
to the line 𝛳 = 0 and directed
to the right. Moreover, r = a - b
cos𝛳 is directed to the left.
r = a - b cosϴ
The graph of the equation
r = a + b sin𝛳 is a limaçon
that is symmetric with respect
to the line 𝛳 =
𝜋
2
and directed
upward. Moreover r = a - b
sin𝛳 is directed downward
r = a + b sin𝛳
r = a - b sin𝛳
The graph of r = a + b cos𝛳
for various values of a and b.
What
kind
of
Limacon Ans. Cardioid, symmetric with respect
graph is r = 3 – 3 cos𝛳.
What
kind
of
to the line 𝛳 = 0 and directed to the left.
Limacon Ans. Limacon with an inner loop,
graph is r = 5 + 6 sin𝛳?
symmetric with respect to the line 𝛳 =
and directed upward.
Polar equations of rose
curves
The
graphs
of
the
equations r = a cos n𝛳 is a
rose curve that is symmetric
with respect to the line 𝛳 = 0
𝜋
2
The
graphs
of
the
equations r = a sin n𝛳 is a
rose curve that is symmetric
with
respect
to
the
line
𝜋
𝛳 = 2.
When n is an even number,
the number of petals is 2n.
When n is an odd number,
the number of petals is n.
How many petals are in the Ans.
graph of
a. n=3 is odd, 3 petals
a. r = 4 cos 3𝛳?
b. n=2 is even, 2n = 4 petals
b. r = 5 sin 2𝛳?
Polar
equations
of
Lemniscates
The graphs of the polar
equation 𝑟 2 = 𝑎2 sin 2𝛳 is a
lemniscate
symmetric
respect to the line 𝛳 =
with
𝜋
4
and
the pole.
The graphs of the polar
equation 𝑟 2 = 𝑎2 cos 2𝛳 is a
lemniscate
symmetric
with
respect to the line 𝛳 =0, the
line 𝛳 =
𝜋
2
, and the pole.
Their
graphs
characteristic
have
a
shape
consisting of two loops that
intersect at the pole.
Graph 𝑟 2 = 42 cos 2
Are there any points on the
2
graph of 𝑟 = 9 cos 2𝛳 for
𝛳<
3𝜋
4
𝜋
4
<
?
Ans. No points are on the graph of 𝑟 2 =
𝜋
9 cos 2𝛳 for 4 < 𝛳 <
3𝜋
4
. Because if you
find the value of r you will get
r=3√cos 2𝛳. If you substitute the value
< 𝛳<
3𝜋
4
𝜋
4
you will get negative value of
cos 2𝛳, and negative inside square root
is imaginary. For example, the value of
theta is 90 degrees and
cos(2x90°) = -1.
F. Fixing Skills
To
enhance
knowledge
your
about
graphing
equations
in
polar
coordinates,
answer
he
following:
1.
Sketch
and
explain
Answer:
how to graph r = 2 - 2 sin 𝛳.
From the general equation of a
Limacon r = a + b sin𝛳 with |a| = |b|
(|2| = |2|), the graph of r = 2 - 2 sin𝛳 is
a cardioid that is symmetric with
respect to the line 𝛳 =
below.
𝜋
2
directed
Because we know that the graph is
heart-shaped, we can sketch the
graph by finding r for a few values of
𝛳.
2. Sketch and explain how
to graph 𝑟 2 = 9 cos 2𝛳.
𝛳
0
𝜋
2
π
3𝜋
2
r
2
0
2
4
Answer:
The equation 𝑟 2 = 9 cos 2𝛳 is of the
form 𝑟 2 = 𝑎2 cos 2𝛳 with 𝑎 = 3. Thus,
the graph is a lemniscate that is
symmetric with respect to the line
𝜋
𝛳 = 0, the line 𝛳 = 2 , and the pole.
Start by plotting a few points in the
𝜋
interval 0 < u < 4 .
𝛳
0
r
±3
𝜋
12
𝜋
6
±2.79 ±2.12
𝜋
4
0
The complete graph can now be
produced by using symmetry with
respect to the line 𝛳 = 0 or by using
symmetry with respect to the line
𝜋
𝛳 = 2.
3. Generalization
What is polar equation and
different types of curves.
A polar equation is an equation
in ordered pair (r, 𝛳). The graph
of a polar equation is the set of all
points whose ordered pairs are
solutions of the equation.
In graphing polar equations in
polar coordinates, there will be
different curves can be drawn.
Few of these are Circle, Limacon,
Rose Curves, and Lemniscates.
IV.
Evaluation
Part I: Multiple Choice
Direction: Select the best answer to the following multiple-choice questions about
graphing equations in polar coordinates. (1pts each)
1. What is the equation?
a.
b.
c.
d.
r = 4sin(Θ)
r = -2sin(Θ)
r = 2cos(Θ)
r = -4cos(Θ)
2. What is the equation?
a.
b.
c.
d.
3.
a.
b.
c.
d.
Θ = -3π/4
Θ = 5π/4
Θ = -5π/4
Θ = π/4
This particular equation generates what graph
4.
a.
b.
c.
d.
Which equation represents the polar graph?
r = cos 4θ
r = 6 cos 4θ
r = sin 4θ
r = 6 sin 2θ
5.
a.
b.
c.
d.
Which equation represents the polar graph?
r = 6 + 6 sin θ
r = 4 + 5 sin θ
r = 4 - 3 sin θ
r = 12 sin θ
6. Identify the polar equation: r² = 9cos(2θ)
a.
b.
c.
d.
cardioid
lemniscate
rose
limacon with inner loop
7.
a.
b.
c.
d.
8.
Which equation below generates this particular graph?
r = 5sin(4θ)
r = 4cos(5θ)
r = 5cos(4θ)
r = 5cos(8θ)
Determine the polar equation of the given graph.
a.
b.
c.
d.
r² = -9cos(2θ)
r² = 9cos(2θ)
r² = -9sin(2θ)
r² = 9sin(2θ)
9. r θ = 3 -3 sin(θ) Describe the curve
a.
b.
c.
d.
Inner loop Limacon
Circle
Rose with 3 leaves
Cardioid
10. The polar coordinate system is based on a point, called the ______ and a ray,
called the _______ axis.
a.
b.
c.
d.
e.
pole, x
radias, polar
radias, x
pole, polar
Test II: Sketch the graph of r = 2 sin 3𝛳. Show your solution. (10pts)
Answer Key:
Part I: Multiple Choice
1. a
5. a
9. d
2. a
6. d
10. d
3. a
7. b
4. a
8. d
Test II: Sketch the graph of r = 2 sin 3𝛳. Show your solution. (10pts)
From the general equation of a rose curve of r = a sin 𝑛𝛳, with a = 2 and n = 3,
𝜋
the graph of r = 2 sin 3𝛳 is a rose curve that is symmetric with respect to the line 𝛳 = 2 .
Because n is an odd number (n = 3), there will be three petals in the graph.
𝛳
0
𝜋
18
𝜋
6
5𝜋
18
𝜋
3
7𝜋
18
𝜋
2
r
0
1
2
1
0
-1
-2
V.
Assignment
Write your answer on a one whole sheet of pad paper.
A. Sketch the graph of the following polar equations;
1. 8 cos 𝛳+ r = 5
2. -cos 2𝛳 + 𝑟 2 = 22
Answer key.
1. 8 cos 𝛳+ r = 5

Transform the equation into r = a ± b cos𝛳,
8 cos 𝛳+ r = 5
r = 5 – 8 cos 𝛳

𝑎
From the general equation of a limaçon r = a - b cos 𝛳 with |𝑏 | < 1, the graph of r
5
= 5 - 8 cos𝛳 with |−8 | < 1 is a Limacon with an inner loop that is symmetric with
respect to the line 𝛳 = 0 directed to left.
Because we know that the graph is with an inner loop, we can sketch the graph by
finding r for a few values of 𝛳.
𝛳
0
𝜋
6
𝜋
3
𝜋
2
2𝜋
3
5𝜋
6
π
7𝜋
6
4𝜋
3
3𝜋
2
r
-3
-1.93
1
5
9
11.93
13
11.93
9
5
𝑟
2. (2) 2= cos 2𝛳
 Transform the equation into 𝑟 2 = 𝑎2 cos 2𝛳
𝑟
(2) 2= cos 2𝛳
𝑟2
22
= cos 2𝛳
𝑟 2 = 22 cos 2𝛳
The equation 𝑟 2 = 22 cos 2𝛳 is of the form 𝑟 2 = 𝑎2 cos 2𝛳 with 𝑎 = 2. Thus, the
graph is a lemniscate that is symmetric with respect to the line
𝜋
𝛳 = 0, the line
𝜋
𝛳 = 2 , and the pole. Start by plotting a few points in the interval 0 < u < 4 .
𝛳
0
r
±2
𝜋
12
𝜋
6
±1.86 ±1.41
𝜋
4
0
The complete graph can now be produced by using symmetry with respect to the
line 𝛳 = 0 or by using symmetry with respect to the line
𝜋
𝛳 = 2.