<< Chapter < Page Chapter >> Page >

Finding zeros and maximum values for a polar equation

Using the equation in [link] , find the zeros and maximum | r | and, if necessary, the polar axis intercepts of r = 2 sin θ .

To find the zeros, set r equal to zero and solve for θ .

2 sin θ = 0 sin θ = 0 θ = sin 1 0 θ = n π where  n  is an integer

Substitute any one of the θ values into the equation. We will use 0.

r = 2 sin ( 0 ) r = 0

The points ( 0 , 0 ) and ( 0 , ± n π ) are the zeros of the equation. They all coincide, so only one point is visible on the graph. This point is also the only polar axis intercept.

To find the maximum value of the equation, look at the maximum value of the trigonometric function sin θ , which occurs when θ = π 2 ± 2 k π resulting in sin ( π 2 ) = 1. Substitute π 2 for θ.

r = 2 sin ( π 2 ) r = 2 ( 1 ) r = 2
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Without converting to Cartesian coordinates, test the given equation for symmetry and find the zeros and maximum values of | r | : r = 3 cos θ .

Tests will reveal symmetry about the polar axis. The zero is ( 0 , π 2 ) , and the maximum value is ( 3 , 0 ) .

Got questions? Get instant answers now!

Investigating circles

Now we have seen the equation of a circle in the polar coordinate system. In the last two examples, the same equation was used to illustrate the properties of symmetry and demonstrate how to find the zeros, maximum values, and plotted points that produced the graphs. However, the circle is only one of many shapes in the set of polar curves.

There are five classic polar curves : cardioids , limaҫons, lemniscates, rose curves , and Archimedes’ spirals . We will briefly touch on the polar formulas for the circle before moving on to the classic curves and their variations.

Formulas for the equation of a circle

Some of the formulas that produce the graph of a circle in polar coordinates are given by r = a cos θ and r = a sin θ , where a is the diameter of the circle or the distance from the pole to the farthest point on the circumference. The radius is | a | 2 , or one-half the diameter. For r = a cos θ ,  the center is ( a 2 , 0 ) . For r = a sin θ , the center is ( a 2 , π ) . [link] shows the graphs of these four circles.

Four graphs side by side. All have radius absolute value of a / 2. First is r=acos(theta), a>0. The center is at (a/2,0). Second is r=acos(theta), a<0. The center is at (a/2,0).  Third is r=asin(theta), a>0. The center is at (a/2, pi). Fourth is r=asin(theta), a<0. The center is at (a/2, pi).

Sketching the graph of a polar equation for a circle

Sketch the graph of r = 4 cos θ .

First, testing the equation for symmetry, we find that the graph is symmetric about the polar axis. Next, we find the zeros    and maximum | r | for r = 4 cos θ . First, set r = 0 , and solve for θ . Thus, a zero occurs at θ = π 2 ± k π . A key point to plot is ( 0 , π 2 ) .

To find the maximum value of r , note that the maximum value of the cosine function is 1 when θ = 0 ± 2 k π . Substitute θ = 0 into the equation:

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

The maximum value of the equation is 4. A key point to plot is ( 4 , 0 ) .

As r = 4 cos θ is symmetric with respect to the polar axis, we only need to calculate r -values for θ over the interval [ 0 , π ] . Points in the upper quadrant can then be reflected to the lower quadrant. Make a table of values similar to [link] . The graph is shown in [link] .

θ 0 π 6 π 4 π 3 π 2 2 π 3 3 π 4 5 π 6 π
r 4 3.46 2.83 2 0 −2 −2.83 −3.46 4
Graph of 4=4cos(theta) in polar coordinates. Points (0, pi/2), (-2, 2pi/3), (4,0), and (2, pi/3) are marked on the circumference.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Investigating cardioids

While translating from polar coordinates to Cartesian coordinates may seem simpler in some instances, graphing the classic curves is actually less complicated in the polar system. The next curve is called a cardioid, as it resembles a heart. This shape is often included with the family of curves called limaçons, but here we will discuss the cardioid on its own.

Questions & Answers

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
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
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
What is algebra
Pearl Reply
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
lolllll if you say so
Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
oh woww I understand
Miranda
haha. already finished college
Jeffrey
how about you? what grade are you now?
Jeffrey
I'm going to 11grade
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
so you better
Miranda
what is the solution of the given equation?
Nelson Reply
which equation
Miranda
I dont know. lol
Jeffrey
please where is the equation
Miranda
ask nelson. lol
Jeffrey
Practice Key Terms 9

Get the best Algebra and trigonometry course in your pocket!





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