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Determine the symmetry of the graph determined by the equation r = 2 cos ( 3 θ ) and create a graph.

Symmetric with respect to the polar axis.
A three-petaled rose is graphed with equation r = 2 cos(3θ). Each petal starts at the origin and reaches a maximum distance from the origin of 2.

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Key concepts

  • The polar coordinate system provides an alternative way to locate points in the plane.
  • Convert points between rectangular and polar coordinates using the formulas
    x = r cos θ and y = r sin θ

    and
    r = x 2 + y 2 and tan θ = y x .
  • To sketch a polar curve from a given polar function, make a table of values and take advantage of periodic properties.
  • Use the conversion formulas to convert equations between rectangular and polar coordinates.
  • Identify symmetry in polar curves, which can occur through the pole, the horizontal axis, or the vertical axis.

In the following exercises, plot the point whose polar coordinates are given by first constructing the angle θ and then marking off the distance r along the ray.

For the following exercises, consider the polar graph below. Give two sets of polar coordinates for each point.

The polar coordinate plane is divided into 12 pies. Point A is drawn on the first circle on the first spoke above the θ = 0 line in the first quadrant. Point B is drawn in the fourth quadrant on the third circle and the second spoke below the θ = 0 line. Point C is drawn on the θ = π line on the third circle. Point D is drawn on the fourth circle on the first spoke below the θ = π line.

Coordinates of point A .

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Coordinates of point B .

B ( 3 , π 3 ) B ( −3 , 2 π 3 )

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Coordinates of point C .

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Coordinates of point D .

D ( 5 , 7 π 6 ) D ( −5 , π 6 )

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For the following exercises, the rectangular coordinates of a point are given. Find two sets of polar coordinates for the point in ( 0 , 2 π ] . Round to three decimal places.

( 3 , −4 ) (3, −4)

( 5 , −0.927 ) ( −5 , −0.927 + π )

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( −6 , 8 )

( 10 , −0.927 ) ( −10 , −0.927 + π )

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( 3 , 3 )

( 2 3 , −0.524 ) ( −2 3 , −0.524 + π )

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For the following exercises, find rectangular coordinates for the given point in polar coordinates.

( −2 , π 6 )

( 3 , −1 )

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( 1 , 7 π 6 )

( 3 2 , −1 2 )

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For the following exercises, determine whether the graphs of the polar equation are symmetric with respect to the x -axis, the y -axis, or the origin.

r = 3 sin ( 2 θ )

Symmetry with respect to the x -axis, y -axis, and origin.

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r = cos ( θ 5 )

Symmetric with respect to x -axis only.

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r = 1 + cos θ

Symmetry with respect to x -axis only.

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For the following exercises, describe the graph of each polar equation. Confirm each description by converting into a rectangular equation.

For the following exercises, convert the rectangular equation to polar form and sketch its graph.

x 2 y 2 = 16

Hyperbola; polar form r 2 cos ( 2 θ ) = 16 or r 2 = 16 sec θ .
A hyperbola with vertices at (−4, 0) and (4, 0), the first pointing out into quadrants II and III and the second pointing out into quadrants I and IV.

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For the following exercises, convert the rectangular equation to polar form and sketch its graph.

3 x y = 2

r = 2 3 cos θ sin θ
A straight line with slope 3 and y intercept −2.

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For the following exercises, convert the polar equation to rectangular form and sketch its graph.

r = 4 sin θ

x 2 + y 2 = 4 y
A circle of radius 2 with center at (2, π/2).

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r = θ

x tan x 2 + y 2 = y
A spiral starting at the origin and crossing θ = π/2 between 1 and 2, θ = π between 3 and 4, θ = 3π/2 between 4 and 5, θ = 0 between 6 and 7, θ = π/2 between 7 and 8, and θ = π between 9 and 10.

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For the following exercises, sketch a graph of the polar equation and identify any symmetry.

r = 1 + sin θ


A cardioid with the upper heart part at the origin and the rest of the cardioid oriented up.
y -axis symmetry

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r = 2 2 sin θ


A cardioid with the upper heart part at the origin and the rest of the cardioid oriented down.
y -axis symmetry

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r = 3 cos ( 2 θ )


A rose with four petals that reach their furthest extent from the origin at θ = 0, π/2, π, and 3π/2.
x - and y -axis symmetry and symmetry about the pole

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r = 2 cos ( 3 θ )


A rose with three petals that reach their furthest extent from the origin at θ = 0, 2π/3, and 4π/3.
x -axis symmetry

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r 2 = 4 cos ( 2 θ )


The infinity symbol with the crossing point at the origin and with the furthest extent of the two petals being at θ = 0 and π.
x - and y -axis symmetry and symmetry about the pole

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[T] The graph of r = 2 cos ( 2 θ ) sec ( θ ) . is called a strophoid. Use a graphing utility to sketch the graph, and, from the graph, determine the asymptote.

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[T] Use a graphing utility and sketch the graph of r = 6 2 sin θ 3 cos θ .


A line that crosses the y axis at roughly 3 and has slope roughly 3/2.
a line

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[T] Use a graphing utility to graph r = 1 1 cos θ .

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[T] Use technology to graph r = e sin ( θ ) 2 cos ( 4 θ ) .


A geometric shape that resembles a butterfly with larger wings in the first and second quadrants, smaller wings in the third and fourth quadrants, a body along the θ = π/2 line and legs along the θ = 0 and π lines.

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[T] Use technology to plot r = sin ( 3 θ 7 ) (use the interval 0 θ 14 π ) .

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Without using technology, sketch the polar curve θ = 2 π 3 .


A line with θ = 120°.

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[T] Use a graphing utility to plot r = θ sin θ for π θ π .

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[T] Use technology to plot r = e −0.1 θ for −10 θ 10 .


A spiral that starts in the third quadrant.

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[T] There is a curve known as the “ Black Hole .” Use technology to plot r = e −0.01 θ for −100 θ 100 .

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[T] Use the results of the preceding two problems to explore the graphs of r = e −0.001 θ and r = e −0.0001 θ for | θ | > 100 .

Answers vary. One possibility is the spiral lines become closer together and the total number of spirals increases.

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Source:  OpenStax, Calculus volume 2. OpenStax CNX. Feb 05, 2016 Download for free at http://cnx.org/content/col11965/1.2
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