<< Chapter < Page Chapter >> Page >

Sketching the graph of a rose curve ( n Even)

Sketch the graph of r = 2 cos 4 θ .

Testing for symmetry, we find again that the symmetry tests do not tell the whole story. The graph is not only symmetric with respect to the polar axis, but also with respect to the line θ = π 2 and the pole.

Now we will find the zeros. First make the substitution u = 4 θ .

0 = 2 cos 4 θ 0 = cos 4 θ 0 = cos u cos 1 0 = u u = π 2 4 θ = π 2 θ = π 8

The zero is θ = π 8 . The point ( 0 , π 8 ) is on the curve.

Next, we find the maximum | r | . We know that the maximum value of cos u = 1 when θ = 0. Thus,

r = 2 cos ( 4 0 ) r = 2 cos ( 0 ) r = 2 ( 1 ) = 2

The point ( 2 , 0 ) is on the curve.

The graph of the rose curve has unique properties, which are revealed in [link] .

θ 0 π 8 π 4 3 π 8 π 2 5 π 8 3 π 4
r 2 0 −2 0 2 0 −2

As r = 0 when θ = π 8 , it makes sense to divide values in the table by π 8 units. A definite pattern emerges. Look at the range of r -values: 2, 0, −2, 0, 2, 0, −2, and so on. This represents the development of the curve one petal at a time. Starting at r = 0 , each petal extends out a distance of r = 2 , and then turns back to zero 2 n times for a total of eight petals. See the graph in [link] .

Sketch of rose curve r=2*cos(4 theta). Goes out distance of 2 for each petal 2n times (here 2*4=8 times).
Rose curve, n even
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Sketch the graph of r = 4 sin ( 2 θ ) .

The graph is a rose curve, n even
Graph of rose curve r=4 sin(2 theta). Even - four petals equally spaced, each of length 4.

Got questions? Get instant answers now!

Sketching the graph of a rose curve ( n Odd)

Sketch the graph of r = 2 sin ( 5 θ ) .

The graph of the equation shows symmetry with respect to the line θ = π 2 . Next, find the zeros and maximum. We will want to make the substitution u = 5 θ .

0 = 2 sin ( 5 θ ) 0 = sin u sin 1 0 = 0 u = 0 5 θ = 0 θ = 0

The maximum value is calculated at the angle where sin θ is a maximum. Therefore,

r = 2 sin ( 5 π 2 ) r = 2 ( 1 ) = 2

Thus, the maximum value of the polar equation is 2. This is the length of each petal. As the curve for n odd yields the same number of petals as n , there will be five petals on the graph. See [link] .

Create a table of values similar to [link] .

θ 0 π 6 π 3 π 2 2 π 3 5 π 6 π
r 0 1 −1.73 2 −1.73 1 0
Graph of rose curve r=2sin(5theta). Five petals equally spaced around origin. Point (2, pi/2) on edge is marked.
Rose curve, n odd
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Sketch the graph of r = 3 cos ( 3 θ ).

Graph of rose curve r=3cos(3theta). Three petals equally spaced from origin.

Rose curve, n odd

Got questions? Get instant answers now!

Investigating the archimedes’ spiral

The final polar equation we will discuss is the Archimedes’ spiral, named for its discoverer, the Greek mathematician Archimedes (c. 287 BCE - c. 212 BCE), who is credited with numerous discoveries in the fields of geometry and mechanics.

Archimedes’ spiral

The formula that generates the graph of the Archimedes’ spiral    is given by r = θ for θ 0. As θ increases, r increases at a constant rate in an ever-widening, never-ending, spiraling path. See [link] .

Two graphs side by side of Archimedes' spiral. (A) is r= theta, [0, 2pi]. (B) is r=theta, [0, 4pi]. Both start at origin and spiral out counterclockwise. The second has two spirals out while the first has one.

Given an Archimedes’ spiral over [ 0 , 2 π ] , sketch the graph.

  1. Make a table of values for r and θ over the given domain.
  2. Plot the points and sketch the graph.

Sketching the graph of an archimedes’ spiral

Sketch the graph of r = θ over [ 0 , 2 π ] .

As r is equal to θ , the plot of the Archimedes’ spiral begins at the pole at the point (0, 0). While the graph hints of symmetry, there is no formal symmetry with regard to passing the symmetry tests. Further, there is no maximum value, unless the domain is restricted.

Create a table such as [link] .

θ π 4 π 2 π 3 π 2 7 π 4 2 π
r 0.785 1.57 3.14 4.71 5.50 6.28

Notice that the r -values are just the decimal form of the angle measured in radians. We can see them on a graph in [link] .

Graph of Archimedes' spiral r=theta over [0,2pi]. Starts at origin and spirals out in one loop counterclockwise. Points (pi/4, pi/4), (pi/2,pi/2), (pi,pi), (5pi/4, 5pi/4), (7pi/4, pi/4), and (2pi, 2pi) are marked.
Archimedes’ spiral
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

(1+cosA+IsinA)(1+cosB+isinB)/(cos@+isin@)(cos$+isin$)
Ajay Reply
hatdog
Mark
how we can draw three triangles of distinctly different shapes. All the angles will be cutt off each triangle and placed side by side with vertices touching
Shahid Reply
bsc F. y algebra and trigonometry pepper 2
Aditi Reply
given that x= 3/5 find sin 3x
Adamu Reply
4
DB
remove any signs and collect terms of -2(8a-3b-c)
Joeval Reply
-16a+6b+2c
Will
is that a real answer
Joeval
(x2-2x+8)-4(x2-3x+5)
Ayush Reply
sorry
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
Y
master
X2-2X+8-4X2+12X-20=0 (X2-4X2)+(-2X+12X)+(-20+8)= 0 -3X2+10X-12=0 3X2-10X+12=0 Use quadratic formula To find the answer answer (5±Root11i)/3
master
Soo sorry (5±Root11* i)/3
master
x2-2x+8-4x2+12x-20 x2-4x2-2x+12x+8-20 -3x2+10x-12 now you can find the answer using quadratic
Mukhtar
2x²-6x+1=0
Ife
explain and give four example of hyperbolic function
Lukman Reply
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
Racelle Reply
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Anurag Reply
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
y2=4ax= y=4ax/2. y=2ax
akash
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
Jhovie Reply
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
thanks
Jhovie
What do you call a relation where each element in the domain is related to only one value in the range by some rules?
Charmaine Reply
A banana.
Yaona
given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
Snalo Reply
what are you up to?
Mark Reply
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
aap konsi country se ho
jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
Propessor Reply
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
sita Reply
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
Practice Key Terms 9

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask