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How are the polar axes different from the x - and y -axes of the Cartesian plane?

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Explain how polar coordinates are graphed.

Determine θ for the point, then move r units from the pole to plot the point. If r is negative, move r units from the pole in the opposite direction but along the same angle. The point is a distance of r away from the origin at an angle of θ from the polar axis.

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How are the points ( 3 , π 2 ) and ( 3 , π 2 ) related?

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Explain why the points ( 3 , π 2 ) and ( 3 , π 2 ) are the same.

The point ( 3 , π 2 ) has a positive angle but a negative radius and is plotted by moving to an angle of π 2 and then moving 3 units in the negative direction. This places the point 3 units down the negative y -axis. The point ( 3 , π 2 ) has a negative angle and a positive radius and is plotted by first moving to an angle of π 2 and then moving 3 units down, which is the positive direction for a negative angle. The point is also 3 units down the negative y -axis.

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Algebraic

For the following exercises, convert the given polar coordinates to Cartesian coordinates with r > 0 and 0 θ 2 π . Remember to consider the quadrant in which the given point is located when determining θ for the point.

( 5 , π )

( 5 , 0 )

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( 3 , π 6 )

( 3 3 2 , 3 2 )

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For the following exercises, convert the given Cartesian coordinates to polar coordinates with r > 0 , 0 θ < 2 π . Remember to consider the quadrant in which the given point is located.

( 4 , 2 )

( 2 5 ,   0.464 )

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( 3 , −5 )

( 34 , 5.253 )

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( 8 , 8 )

( 8 2 , π 4 )

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For the following exercises, convert the given Cartesian equation to a polar equation.

y = 2 x 4

r = s i n θ 2 c o s 4 θ 3

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x 2 + y 2 = 3 x

r = 3 cos θ

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x 2 y 2 = 3 y

r = 3 sin θ cos ( 2 θ )

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x 2 = 9 y

r = 9 sin θ cos 2 θ

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9 x y = 1

r = 1 9 cos θ sin θ

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For the following exercises, convert the given polar equation to a Cartesian equation. Write in the standard form of a conic if possible, and identify the conic section represented.

r = 4 cos θ

x 2 + y 2 = 4 x or ( x 2 ) 2 4 + y 2 4 = 1 ; circle

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r = 4 sin θ + 7 cos θ

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r = 6 cos θ + 3 sin θ

3 y + x = 6 ; line

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r = 3 csc θ

y = 3 ; line

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r 2 = 4 sec θ csc θ

x y = 4 ; hyperbola

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r 2 = 4

x 2 + y 2 = 4 ; circle

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r = 1 4 cos θ 3 sin θ

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r = 3 cos θ 5 sin θ

x 5 y = 3 ; line

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Graphical

For the following exercises, find the polar coordinates of the point.

For the following exercises, plot the points.

For the following exercises, convert the equation from rectangular to polar form and graph on the polar axis.

5 x y = 6

r = 6 5 cos θ sin θ

Plot of given line in the polar coordinate grid
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x 2 + ( y 1 ) 2 = 1

r = 2 sin θ

Plot of given circle in the polar coordinate grid
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( x + 2 ) 2 + ( y + 3 ) 2 = 13

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x 2 + y 2 = 3 x

r = 3 cos θ

Plot of given circle in the polar coordinate grid.
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For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.

r = 4

x 2 + y 2 = 16

Plot of circle with radius 4 centered at the origin in the rectangular coordinates grid.
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r = −10 sin θ

x 2 + ( y + 5 ) 2 = 25

Plot of circle with radius 5 centered at (0,-5).
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Technology

Use a graphing calculator to find the rectangular coordinates of ( 2 , π 5 ) . Round to the nearest thousandth.

( 1.618 , 1.176 )

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Use a graphing calculator to find the rectangular coordinates of ( 3 , 3 π 7 ) . Round to the nearest thousandth.

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Use a graphing calculator to find the polar coordinates of ( 7 , 8 ) in degrees. Round to the nearest thousandth.

( 10.630 , 131.186° )

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Use a graphing calculator to find the polar coordinates of ( 3 , 4 ) in degrees. Round to the nearest hundredth.

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Use a graphing calculator to find the polar coordinates of ( 2 , 0 ) in radians. Round to the nearest hundredth.

( 2 , 3.14 ) o r ( 2 , π )

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Extensions

Describe the graph of r = a sec θ ; a > 0.

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Describe the graph of r = a sec θ ; a < 0.

A vertical line with a units left of the y -axis. 

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Describe the graph of r = a csc θ ; a > 0.

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Describe the graph of r = a csc θ ; a < 0.

A horizontal line with a units below the x -axis.

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What polar equations will give an oblique line?

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For the following exercise, graph the polar inequality.

θ = π 4 , r 2

Graph of ray starting at (2, pi/4) and extending in a positive direction along pi/4 - polar coordinate grid.
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θ = π 4 , r −3

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0 θ π 3 , r < 2

Graph of the shaded region 0 to pi/3 from r=0 to 2 with the edge not included (dotted line) - polar coordinate grid
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π 6 < θ π 3 , 3 < r < 2

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Questions & Answers

find general solution of the Tanx=-1/root3,secx=2/root3
Nani Reply
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
Sanjay Reply
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
duru Reply
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
Koru Reply
where can I get indices
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I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
about complex fraction
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
nothing. I accidentally press it
Amie
you guys know any app with matrices?
Khay
Ok
Cromwell
Solve the x? x=18+(24-3)=72
Leizel Reply
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Deadra Reply
Need help with this question please
Deadra
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
yah
immy
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
7.5 and 37.5
Nando
how would this look as an equation?
Hayden
5x+x=45
Khay
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
Kenneth
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
Practice Key Terms 3

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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