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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

answer and questions in exercise 11.2 sums
Yp Reply
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cos (-z)= cos z .
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(x+x)3=?
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what is the function of sine with respect of cosine , graphically
Karl Reply
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
Aashish Reply
sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
 Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
I don't understand how radicals works pls
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How look for the general solution of a trig function
collins Reply
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
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sinx sin2x is linearly dependent
cr
root under 3-root under 2 by 5 y square
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The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
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why two x + seven is equal to nineteen.
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The numbers cannot be combined with the x
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because x is 6
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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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