Polar System

Polar Coordinates

Definition. A polar coordinate system in a plane consists of a fixed point O, called the pole or origin, and a ray emanatingfrom the pole, called the polar axis. In such a coordinate system we associate with each point P in the plane a pair of polarcoordinates (r, θ), where r is the directed distance from P to the pole and θ is the measure of the angle from the polar axis

to the ray OP . The number r is called the radial coordinate of P and the number θ the angular coordinate or polar angle of P .

Remark.

(1) The polar axis is usually drawn horizontally and extending to the right.(2) The polar plane is composed of concentric circles with center at the pole and lines passing through the pole.(3) When an angle is measured in the counter-clockwise direction then the angle measure is positive and if an angle

is measured in the clockwise direction then the angle measure is negative.(4) If r > 0 then P (r, θ) is the point r units from the pole and on the terminal side of the angle with measure θ.(5) If r < 0 then P (r, θ) is the point r units from the pole and on the terminal side of the angle with measure θ+ π. The

point is r units from the pole and on the extension of the terminal side of the angle with measure θ.

Example. Plot the following points.

(1) (1, π/6)(2) (−3,345◦)(3) (1, π)

(4) (2,120◦)(5) (−3, π/4)(6) (−2,−5π/4)

(7) (7,225◦)(8) (−5,−2π/3)(9) (−6,−7π/4)

(10) (−4, π/3)(11) (0,−35π/2)(12) (8,−π/2)

0

Π

12

Π

6

Π

4

Π

3

5 Π

12

Π

27 Π

122 Π

3

3 Π

4

5 Π

6

11 Π

12

Π

13 Π

12

7 Π

6

5 Π

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4 Π

3 17 Π

123 Π

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12

5 Π

3

7 Π

4

11 Π

6

23 Π

12

0.

2.

4.

6.

8.

10.

Remark.

(1) Ever pair of polar coordinates (r, θ) is associated with a unique point on the polar coordinate system. However,every point on the polar coordinate system is associated with infinitely many pairs of polar coordinates, i.e., thepolar coordinates of a point are not unique.

(2) The polar coordinates (0, θ) represent the pole for all choices of θ.(3) In general, if a point P has polar coordinates (r, θ), then (r, θ+2nπ) or (−r, θ+(2n+1)π) are also polar coordinates

of P for any integer n.(4) If we restrict r > 0 and 0 ≤ θ < 2π then a point P on the polar coordinate system will have a unique set of polar

coordinates.(5) In problems involving differentiation or integration formulas, angles must be in radians because those formulas

were derived under that assumption. We will use radian measure for polar angles, except in applications wheredegree measure is more convenient and radian measure is not required.

F MATHEMATICS 54 TWHFV AND TWHFW2 F POLAR CURVES

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RELATIONSHIP BETWEEN POLAR AND RECTANGULAR COORDINATES

If we superimpose a rectangular xy-coordinate system on top of the polar coordinate system with the positive x-axiscoinciding with the polar axis, then every point P will have both rectangular coordinates (x, y) and polar coordinates(r, θ). These coordinates are related by the following equations:

r = ±√

x2 + y2, tanθ =y

xx = r cosθ, y = r sin θ

Example. Find the rectangular coordinates of the points whose polar coordinates are given.

(1) (6,2π/3)(2) (4, π/3)

(3) (2,−π/6)(4) (3,5π/4)

(5) (−2, π/2)

(6) (√20,−π/4)

(7) (−2,−5π/6)(8) (−1,7π/4)

(9) (−1,2π)(10) (−2,15π/2)

Example. Find polar coordinates of the points whose rectangular coordinates are given. Assume r > 0 and 0 ≤ θ < 2π.

(1) (−2,2√3)

(2) (1,−1)(3) (−1,−

√3)

(4) (−3,4)(5) (−

√3,1)

(6) (−1,−1)(7) (2,2

√3)

(8) (3,−√3)

(9) (−1/5,1/5)(10) (1,0)

GRAPHS IN POLAR COORDINATES

In a polar coordinate system, points have infinitely many different pairs of polar coordinates, so that a given point mayhave some polar coordinates that satisfy an equation and others that do not. Given an equation in r and θ, we defineits graph in polar coordinates to consist of all points with at least one pair of coordinates (r, θ) that satisfy the equation.Meanwhile, a polar equation of a graph is an equation that is satisfied by at least one pair of polar coordinates of each pointof the graph.

Remark. A graph may have more than one polar equation.

Example. Determine if the following points are on the graph of the given polar equation.

(1) r = 1− 2 sinθ• P (−2,5π/6)

(2) r =3

cosθ− 1• P (3, π/2)

(3) r = 2cos3θ• pole• P (1,11π/9)• P (−2,−2π/3)

Example. Express the given equations in polar coordinates.

(1) x = 3(2) y2 = 2x

(3) x2 + y2 = 9(4) 9xy = 4

(5) 2x2 + 3y2 = 4(6) x2(x2 + y2) = y2

Example. Transform the given polar equations to rectangular coordinates.

(1) θ = π/2(2) r = 1

(3) r = 3cosθ(4) r2 = 9cos2θ

(5) r = a cosθ+ b sinθ

(6) r =1

2+ 3 sinθ

Example. Sketch the graphs of the following in polar coordinates.

(1) r = 1 (2) θ = π/4 (3) r = sinθ

Families of lines and rays through the pole. The equation θ = θ0 where θ0 ∈ R represents the line that passes through thepole and makes an angle of θ0 with the polar axis, i.e., the line has slope equal to tanθ0. If r is restricted to be nonnegative,then the graph of the equation θ = θ0 is the ray that emanates from the pole and makes an angle of θ0 with the polar axis.Thus, as θ0 varies, the equation θ = θ0 produces either a family of lines through the pole or a family of rays through thepole, depending on the restrictions on r.

Example. Sketch the graph of the following polar equations:

(1) θ = π/3 (2) θ = 5π/6

Remark.

(1) An equation of the polar axis is θ = 0 and an equation of the π/2-axis is θ = π/2.(2) The graph of the polar equation θ = θ0 is the same as the graphs of the equations θ = θ0 ± nπ where n ∈ Z. For

example, the line with polar equation θ = π/3 can also have equations θ = 4π/3,−2π/3, etc.

The graph of r cosθ = a is a line parallel to the π/2-axis and intersecting the polar axis at the point with polar coordinate(a,0). Similarly, the graph of r sin θ = b is a line parallel to the polar axis and intersecting the π/2-axis at the point withpolar coordinates (b, π/2).


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Remark.

(1) The line with polar equation r cosθ = a is to the right of the π/2-axis if a > 0 and to the left of the π/2 axis if a < 0.(2) The line with polar equation r sin θ = b is above the polar axis if b > 0 and below the polar axis if b < 0.

Example. Sketch the graph of the following polar equations.

(1) r cosθ = 1 (2) r sin θ = −3

The graph of m(r cos θ)− r sin θ+ b = 0 is a line with slope m passing through (b, π/2) and (−b/m,0).

Example. Sketch the graph of the equation 2 cosθ− 2 sinθ = 1.

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r cosθ = a, a < 0 r cosθ = a, a > 0 θ = θ0 r sinθ = b, b < 0 r sin θ = b, b > 0

Families of Circles. The graph of the polar equation r = a where a ∈ R is a circle with center at the pole and radius oflength |a|.Example. Sketch the graph of the following polar equations.

(1) r = 3 (2) r = −2

Remark. The graph of the polar equation r = a can also have polar equation r = −a.

The graph of the polar equation r = 2a cosθ is a circle with center on the polar axis or its extension (point with polarcoordinates (a,0)), with radius |a|, passing through the pole and tangent to the π/2-axis. Similarly, the graph of the polarequation r = 2b sinθ is a circle with center on the π/2-axis or its extension (point with polar coordinates (b, π/2)), withradius |b|, passing through the pole and tangent to the polar axis.


(1) r = 3cosθ (2) r = −2 sinθ

Remark.

(1) The circle with polar equation r = 2a cosθ is to the right of the π/2-axis if a > 0 and to the left of the π/2-axis ifa < 0.

(2) The circle with polar equation r = 2b sinθ is above the polar axis if b > 0 and below the polar axis if b < 0.

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r = 2a cosθ, a > 0 r = 2a cosθ, a < 0 r = a r = 2a sinθ, a > 0 r = 2a sinθ, a < 0

Symmetry Tests.

Theorem. A curve in polar coordinates is symmetric about the

(1) polar axis if an equivalent equation is obtained when (r, θ) is replaced by either (r,−θ) or (−r, π− θ).(2) π/2-axis if an equivalent equation is obtained when (r, θ) is replaced by either (r, π− θ) or (−r,−θ).(3) pole if an equivalent equation is obtained when (r, θ) is replaced by either (r, π+ θ) or (−r, θ).

Remark. The converse of each part of the theorem is false.

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r = 2− sin(θ/2), 0 ≤ θ ≤ 4π r = cos(θ/4), 0 ≤ θ ≤ 8π

Remark. A graph that is symmetric about both the polar axis and the π/2-axis is also symmetric about the origin.


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Example. Determine if the graph of the following polar equations is symmetric with respect to the polar axis, π/2-axisand pole.

(1) r2 = 4cos2θ (2) r = 2sin3θ

Remark.

(1) If the polar equation r = f(θ) is(a) a function of cosθ only, then the graph of r = f(θ) is symmetric with respect to the polar axis.(b) a function of sinθ only, then the graph of r = f(θ) is symmetric with respect to the π/2-axis.

(2) The graph of the polar equation r2n = f(θ) where n ∈ Z is symmetric with respect to the pole.

Observe that we can define the graph of a polar equation by a pair of parametric equations:

Theorem. The graph of the polar equation r = f(θ) is defined by the parametric equations x = f(θ) cosθ and y = f(θ) sinθ.

Example. Express the polar equation r = 2+ sin10θ− π

4parametrically.

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2

TANGENT LINES TO POLAR CURVES

Theorem. Let r be a differentiable function of θ. If dx/dθ and dy/dθ are continuous and dx/dθ 6= 0, then y is a differentiablefunction of x, and

dy

dx=

sin θdr

dθ+ r cosθ

cosθdr

dθ− r sin θ

.

Corollary. If m is the slope of the tangent line to the graph of r = f(θ) at the point P (r, θ) then m =dr/dθ tan θ+ r

dr/dθ− r tan θif cosθ 6= 0

and m = −1

r

dr

dθif cosθ = 0.

Remark. If the polar curve r = f(θ) passes through the pole at θ = θ0, then the slope of the tangent line to the graph ofr = f(θ) at the pole is m= tan θ0. Hence, the tangent line to the graph of r = f(θ) at the pole is the line with polar equationθ = θ0.

Example. Answer the following completely.

(1) Find the slope of the tangent line to the circle r = 4cosθ at the point where θ = π/4.(2) Find the slope of the tangent line to the graph of r = 4sin3θ at the point where θ = π/6 and find the equation of

the tangent line at the pole.(3) Find the slope of the tangent line to the graph of r = 1− 2 cosθ at the pole where θ = 3π/2.

Exercise. Find the Cartesian equation of the tangent line to the graph of r = 3cos2θ at the point with polar coordinates(3/2, π/6).

Theorem. Let β be the radian measure of the angle between the line OP and the tangent line to the graph of r = f(θ) at the point

P (r, θ) where P is not on the π/2-axis or its extension. If angle β is measured from OP counterclockwise to the tangent line and

0 ≤ β < π then tanβ = r/dr

dθ.

Example. Consider the polar curve r = 2+ 2 sinθ. Find the angle measures α and β described above where θ = π/6.

Exercise. Use angle β to find the slope of the tangent line to the graph of r = sinθ tanθ at the point with polar coordinates

(√2/2, π/4) in rectangular coordinates.

Suggestions on How to Proceed in Sketching the Graph of a Polar Equation.

(1) Test the graph of the polar equation for symmetry with respect to the polar axis, π/2-axis and the pole.(2) Find the polar intercepts and the π/2 intercepts of the polar equation.

(a) To find the polar intercepts of the polar equation, we let θ = 0 and θ = π.(b) To find the π/2 intercepts of the polar equation, we let θ = π/2 and θ = 3π/2.

(3) Determine the tangent lines at the pole.(a) The equations of the tangent lines at the pole are θ= θ1, θ= θ2, . . . , θ= θk where (0, θ1), (0, θ2), . . . , (0, θk) satisfy

the polar equation.(b) Assign one or two or even three values to θ that are between two consecutive polar tangents and plot corre-

sponding points.(4) Draw a smooth curve connecting the points in the order of increasing θ.


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Families of Cardioids and Limaçons. Equations of the form r = a± b sinθ and r = a± b cosθ in which a > 0 and b > 0represent polar curves called limaçons.

Example. Sketch a graph of the following polar equations.

(1) r = 1+ 2cosθ (2) r = 1− sin θ (3) r = 3− 2 cosθ (4) r = 3+ sin θ

Types of Limaçons. The graph of the polar equation r = a± b cosθ and r = a± b sinθ where a, b > 0 is a

(1) limaçon with a loop if 0 < a/b < 1(2) cardioid (heart-shape) if a/b = 1

(3) limaçon with a dent or dimpled limaçon if 1 < a/b < 2(4) convex limaçon if a/b ≥ 2

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0 < a/b < 1 a/b = 1 1 < a/b < 2 a/b ≥ 2

Remark.

(1) The limaçon with a loop and the cardioid pass through the pole while the limaçon with a dent and the convexlimaçon do not intersect the pole.

(2) A limaçon with a dent has four tangent lines that is perpendicular to its axis of symmetry (either horizontal orvertical) while a convex limaçon has only two.

Proposition. If a, b > 0, then the graphs of

(1) r = a+ b cosθ is symmetric with respect to the polar axis and the limaçon points to the right.(2) r = a− b cosθ is symmetric with respect to the polar axis and the limaçon points to the left.(3) r = a+ b sinθ is symmetric with respect to the π/2 axis and the limaçon points upward.(4) r = a− b sinθ is symmetric with respect to the π/2 axis and the limaçon points downward.

Exercise. Find the points on the cardioid r = 1− cosθ at which there is a horizontal tangent line, a vertical tangent line, ora singular point.

Families of Rose Curves. Equations of the form r = a cosnθ and r = a sinnθ, where a > 0 and n is a positive integerrepresent families of flower-shaped curves called roses.

The rose consists of n equally spaced petals or leaves of radius a if n is odd and 2n equally spaced petals of radius a if nis even. A rose with an even number of petals is traced out exactly once as θ varies over the interval 0 ≤ θ ≤ 2π and a rosewith an odd number of petals is traced out exactly once as θ varies over the interval 0 ≤ θ ≤ π.


(1) r = sin2θ (2) r = 2cos3θ

Remark. A circle can be considered a one-petal rose since if n = 1 we obtain r = a sinθ or r = a cosθ, the graph of which isa circle of radius a/2 passing through the pole and with center at the polar axis or on the π/2-axis.

Exercise. Determine the equations of the tangent line(s) to the rose r = sin3θ at the pole.

r = a cosnθ

r = a sinnθ

n = 2 n = 3 n = 4 n = 5 n = 6

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Lemniscates. The graph of the polar equation of the form r2 = a cos2θ or r2 = a sin2θ is a lemniscate.


(1) r2 = 4cos2θ (2) r2 = 9sin2θ

Remark. The lemniscate with polar equation of the form r2 = a cos2θ is symmetric with respect to the polar axis, π/2-axisand the pole while the lemniscate with polar equation of the form r2 = a sin2θ is symmetric with respect to the pole.

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Families of Spirals. A spiral is a curve that coils around a central point.

Example. Sketch the graph of r = a(1− cosθ) in polar coordinates, assuming a to be a positive constant.

Exercise. Sketch the graph of r = θ, where θ ≥ 0.

Remark. The graph of the polar equation r = θ where θ ≥ 0 is called the Spiral of Archimedes.

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Archimidean Parabolic Hyperbolic Logarithmic

ARC LENGTH OF A POLAR CURVE

Theorem. If no segment of the polar curve r = f(θ) is traced more than once as θ increases from α to β, and if dr/dθ is continuousfor α ≤ θ ≤ β, then the arc length L from θ = α to θ = β is

L =

ˆ β

α

√

[f(θ)]2 + [f ′(θ)]2 dθ =

ˆ β

α

√

r2 +

(

dr

dθ

)2

dθ

Example. Find the total arc length of

(1) the Spiral of Archimedes where 0 ≤ θ ≤ π. (2) the cardioid with equation r = 1+ cosθ.

Exercise. Set-up the integral equal to the length of

(1) the limaçon with equation r = 3− 2 sinθ (2) the inner loop of the limaçon r = 2− 4 cosθ

Exercise. Show that the circumference of a circle of radius a is equal to 2πa by considering the length of arc of the graphof the polar equation r = 2a sinθ.

Intersection of Graphs of Polar Equations. Let P be a point with polar coordinates (r, θ). Then P can also have polar coor-dinates (r, θ + 2nπ) or (−r, θ+ (2n+ 1)π) where n ∈ Z, that is, P can also have polar coordinates ((−1)nr, θ + nπ) wheren ∈ Z.

Theorem. If r = f(θ) is a polar equation of a graph then other polar equations of the graph are of the form (−1)nr = f(θ + nπ)where n ∈ Z.

General Method for finding the coordinates of all points of intersection of the graphs of the polar equations r = f(θ) and r = g(θ):

(1) Test if the pole is a point of intersection.• The pole is on both curves if we set r = 0 and the equations f(θ) = 0 and g(θ) = 0 both have solutions not

necessarily the same.(2) Determine all the distinct polar equations of the graphs of r= f(θ) and r= g(θ) obtaining r= f1(θ), r = f2(θ), . . . , r=

fs(θ) and r = g1(θ), r = g2(θ), . . . , r = gt(θ), respectively.

(3) Solve the systems of polar equations

{

r = fi(θ)

r = gj(θ)for i = 1, . . . , s and j = 1, . . . , t.


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Example. Determine all points of intersection of the following.

(1) r = 2sinθ and r = 2cosθ(2) r = 1 and r22 sin2θ

(3) r = 1 and r = 2cos2θ

AREA IN POLAR COORDINATES

Theorem. Let α and β be angle measures that satisfy the condition α < β ≤ α+ 2π. If the function r = f(θ) is continuous andeither nonnegative or nonpositive for every θ ∈ [α,β], then the area A of the region enclosed by the polar curve r = f(θ) and the linesθ = α and θ = β is

A =1

2

ˆ β

α

[f(θ)]2 dθ =1

2

ˆ β

α

r2 dθ.

Example. Find the exact area of the region

(1) in the first quadrant that is within the cardioid r = 1− cosθ.(2) within the cardioid r = 1− cosθ.(3) enclosed by the rose r = cos2θ.(4) that is inside of the cardioid r = 4+ 4cosθ and outside of the circle r = 6.

Example. Setup the integral that is equal to the area of the

(1) region bounded by the graph of r = 2+ 2 sinθ(2) the inner loop of the graph of r = 2− 4 cosθ

Example. Setup the integral that is equal to the area of the

(1) of the region outside the graph of r = 1 and inside the graph of r2 = 2sin2θ(2) of the region outside the graph of r = 1 and inside the graph of r = 2cos2θ(3) of the region inside the graph of r = 2 and outside the graph of r = 2− 2 cosθ(4) of the region outside the graph of r = 1 but inside the graph of r = 2sinθ(5) of the region inside the graphs of r = 1 and r = 2sinθ(6) of the region enclosed by the graphs of r = 2sin2θ and r = 2cos2θ

Exercise. Setup the integral equal to the area

(1) of the region bounded by the graphs of r = 3+ cosθ

(2) of the region outside the graph of r = 1+ cosθ and inside the graph of r =√3 sinθ

(3) of the region common to the interior of the graphs of r = 4sinθ and r = 4cosθ(4) of the region enclosed by the graphs of r = 2 and r = 3− 2 cosθ(5) of the region outside the inner loop but inside the limaçon r = 1+ 2cosθ

A UNIFIED TREATMENT OF CONIC SECTIONS

Theorem. Suppose that a point P moves in the plane determined by a fixed point and a fixed line, where the focus does not lie onthe directrix. If the point moves in such a way that the ratio of its undirected distance to the focus to its undirected distance to thedirectrix is some positive constant e, then the curve traced by the point is a conic section. Moreover, if e = 1, the conic is a parabola; if0 < e < 1 the conic is an ellipse and if e > 1 the conic is a hyperbola.

Remark.

(1) In the theorem above, the fixed point is called a focus of the conic, the fixed line is called a directrix of the conic ande is called the eccentricity of the conic.

(2) A conic with e 6= 1 is called a central conic (ellipse and hyperbola).

Theorem. Suppose the focus of a conic of eccentricity e is at the pole. If d is the undirected distance of the pole to its correspondingdirectrix that is

(1) perpendicular to the polar axis then a polar equation of the conic is r =ed

1± e cosθwhere the plus sign is taken when the

directrix corresponding to the pole is to the right of the pole and the minus sign is taken when it is to the left of the pole.

(2) parallel to the polar axis then a polar equation of the conic is r =ed

1± e sinθwhere the plus sign is taken when the directrix

corresponding to the pole is above the pole and the minus sign is taken when it is below the pole.

Example. Identify the following conics and sketch their graphs.

(1) r =2

1− cosθ

(2) r =4

1+ cosθ

(3) r =6

2− sin θ

(4) r =12

3+ 6 sinθ

Remark. If r is undefined when θ = θ1 and θ = θ2, θ1, θ2 ∈ [0,2π), then the asymptotes of the hyperbola are the lines passingthrough the center of the hyperbola and parallel to the line with equation θ = θ1 and θ = θ2.


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Exercise. Identify the following conics and sketch their graphs.

(1) r =1

2+ sin θ(2) r =

4

2− 2 sinθ(3) r =

7

2+ 4cosθ(4) r =

6

4− cosθ

Example. Find a polar equation of the conic having a focus at the pole and satisfying the given conditions:

(1) parabola; vertex at V (2,3π/2)(2) e = 0.5; corresponding directrix to the focus at the pole has polar equation r cosθ = 10(3) e = 1; point with polar coordinate (2, π/3) on the conic, vertex is on the extension of the polar axis(4) vertices at points with polar coordinates (1, π/2) and (3, π/2)

Exercise. Find a polar equation of the conic having a focus at the pole and satisfying the given conditions:

(1) parabola; vertex at V (1.5,0)(2) point with polar coordinates (8/3, π/6) on conic, equation of directrix corresponding to focus at pole is r sin θ = −4(3) other focus has polar coordinates (−8, π), a vertex at V (2,0) and d = 3

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-2

-1

1

2

3

4


0

Π

12

Π

6

Π

4

Π

3

5 Π

12

Π

27 Π

122 Π

3

3 Π

4

5 Π

6

11 Π

12

Π

13 Π

12

7 Π

6

5 Π

4

4 Π

3 17 Π

123 Π

2

19 Π

12

5 Π

3

7 Π

4

11 Π

6

23 Π

12

0.

2.

4.

6.

8.

10.

0

Π

12

Π

6

Π

4

Π

3

5 Π

12

Π

27 Π

122 Π

3

3 Π

4

5 Π

6

11 Π

12

Π

13 Π

12

7 Π

6

5 Π

4

4 Π

3 17 Π

123 Π

2

19 Π

12

5 Π

3

7 Π

4

11 Π

6

23 Π

12

0.

2.

4.

6.

8.

10.

0

Π

12

Π

6

Π

4

Π

3

5 Π

12

Π

27 Π

122 Π

3

3 Π

4

5 Π

6

11 Π

12

Π

13 Π

12

7 Π

6

5 Π

4

4 Π

3 17 Π

123 Π

2

19 Π

12

5 Π

3

7 Π

4

11 Π

6

23 Π

12

0.

2.

4.

6.

8.

10.

0

Π

12

Π

6

Π

4

Π

3

5 Π

12

Π

27 Π

122 Π

3

3 Π

4

5 Π

6

11 Π

12

Π

13 Π

12

7 Π

6

5 Π

4

4 Π

3 17 Π

123 Π

2

19 Π

12

5 Π

3

7 Π

4

11 Π

6

23 Π

12

0.

2.

4.

6.

8.

10.

0

Π

12

Π

6

Π

4

Π

3

5 Π

12

Π

27 Π

122 Π

3

3 Π

4

5 Π

6

11 Π

12

Π

13 Π

12

7 Π

6

5 Π

4

4 Π

3 17 Π

123 Π

2

19 Π

12

5 Π

3

7 Π

4

11 Π

6

23 Π

12

0.

2.

4.

6.

8.

10.

0

Π

12

Π

6

Π

4

Π

3

5 Π

12

Π

27 Π

122 Π

3

3 Π

4

5 Π

6

11 Π

12

Π

13 Π

12

7 Π

6

5 Π

4

4 Π

3 17 Π

123 Π

2

19 Π

12

5 Π

3

7 Π

4

11 Π

6

23 Π

12

0.

2.

4.

6.

8.

10.

F MATHEMATICS 54 TWHFV AND TWHFW2 F POLAR PLANES

0

Π

12

Π

6

Π

4

Π

3

5 Π

12

Π

27 Π

122 Π

3

3 Π

4

5 Π

6

11 Π

12

Π

13 Π

12

7 Π

6

5 Π

4

4 Π

3 17 Π

123 Π

2

19 Π

12

5 Π

3

7 Π

4

11 Π

6

23 Π

12

0.

1.

2.

3.

4.

5.

0

Π

12

Π

6

Π

4

Π

3

5 Π

12

Π

27 Π

122 Π

3

3 Π

4

5 Π

6

11 Π

12

Π

13 Π

12

7 Π

6

5 Π

4

4 Π

3 17 Π

123 Π

2

19 Π

12

5 Π

3

7 Π

4

11 Π

6

23 Π

12

0.

1.

2.

3.

4.

5.

0

Π

12

Π

6

Π

4

Π

3

5 Π

12

Π

27 Π

122 Π

3

3 Π

4

5 Π

6

11 Π

12

Π

13 Π

12

7 Π

6

5 Π

4

4 Π

3 17 Π

123 Π

2

19 Π

12

5 Π

3

7 Π

4

11 Π

6

23 Π

12

0.

1.

2.

3.

4.

5.

0

Π

12

Π

6

Π

4

Π

3

5 Π

12

Π

27 Π

122 Π

3

3 Π

4

5 Π

6

11 Π

12

Π

13 Π

12

7 Π

6

5 Π

4

4 Π

3 17 Π

123 Π

2

19 Π

12

5 Π

3

7 Π

4

11 Π

6

23 Π

12

0.

1.

2.

3.

4.

5.

0

Π

12

Π

6

Π

4

Π

3

5 Π

12

Π

27 Π

122 Π

3

3 Π

4

5 Π

6

11 Π

12

Π

13 Π

12

7 Π

6

5 Π

4

4 Π

3 17 Π

123 Π

2

19 Π

12

5 Π

3

7 Π

4

11 Π

6

23 Π

12

0.

1.

2.

3.

4.

5.

0

Π

12

Π

6

Π

4

Π

3

5 Π

12

Π

27 Π

122 Π

3

3 Π

4

5 Π

6

11 Π

12

Π

13 Π

12

7 Π

6

5 Π

4

4 Π

3 17 Π

123 Π

2

19 Π

12

5 Π

3

7 Π

4

11 Π

6

23 Π

12

0.

1.

2.

3.

4.

5.

F MATHEMATICS 54 TWHFV AND TWHFW2 F POLAR PLANES

Polar System

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