<< 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

Find the possible value of 8.5 using moivre's theorem
Reuben Reply
which of these functions is not uniformly cintinuous on (0, 1)? sinx
Pooja Reply
which of these functions is not uniformly continuous on 0,1
Basant Reply
solve this equation by completing the square 3x-4x-7=0
Jamiz Reply
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
what is trigonometry
Jean Reply
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
Rubben Reply
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
I think he drive me safe
Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
Pab Reply
More example of algebra and trigo
Stephen Reply
What is Indices
Yashim Reply
If one side only of a triangle is given is it possible to solve for the unkown two sides?
Felix Reply
cool
Rubben
kya
Khushnama
please I need help in maths
Dayo Reply
Okey tell me, what's your problem is?
Navin
the least possible degree ?
Dejen Reply
(1+cosA)(1-cosA)=sin^2A
BINCY Reply
good
Neha
why I'm sending you solved question
Mirza
Teach me abt the echelon method
Khamis
exact value of cos(π/3-π/4)
Ankit Reply
What is differentiation?
Intakhab Reply
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