Workbook: Polar Coordinates WB1 Practice some points

(2 , π2 ) (−1 , π

3 ) (1 , π ) (−2 , 4 π3 ) (3 , 3π

4 )(1.5 ,−π

2 ) (0.5 , 7π6 ) (0.5 ,−5 π

6 ) (2.5 , 11 π6 ) (−1 ,−π

6 )

x=rcosθ

y=rsinθ

θ=arctan( yx )

r2=x2+ y2

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Workbook: Polar Coordinates

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Workbook: Polar Coordinates

WB2 Find the Polar coordinates of the following points:

a) (5, 9)

b) (5, -12)

c) (−√3 ,−1¿

d) (12, 5)

e) (-3, 0)

f) (2, -2)

g) (0, -4)

h) (-3, -4)

i) (−1 ,√3 )

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Workbook: Polar Coordinates

WB3 Find the Cartesian coordinates of the following points:

a) (10 , 4 π3 )

b) (8 , 2π3 )

c) (10 , π2 )

d) (2 , 56

π)

e) (4 , 54

π )

f) (6 ,−23

π)

g) (5 , π )

h) (3 , 2 )

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Workbook: Polar Coordinates

x=rcosθ θ=arctan( yx )

y=rsinθ r2=x2+ y2

Addition formulaesin ( A ± B )=sin A cosB ± sin B sin A

cos ( A ± B )=cos A cos B∓sin A sin B

WB4 Find a Cartesian equation of each of the following curvesSketch the Cartesian graph (x and y axes)

a) r=5

b) r=6 cosecθ

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Workbook: Polar Coordinates

WB5 Find a Cartesian equation of the following curves

a) r=2+cos2 θ

b) r2=sin2θ

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Workbook: Polar Coordinates

WB6 Find a Cartesian equation of the following curves

a) r=2a cosθ−12

π ≤θ< 12

π

b) r2=a2 sin2 θ 0 ≤ θ<π

c) r=2+2co s2θ 0≤θ<π

d) r=2sin θ 0 ≤ θ<π

e) r=acos2 θ−14

π ≤θ ≤ 14

π

f) r=a co secθ 0<θ<π

g) r= 43+cosθ

0<θ<π

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Workbook: Polar Coordinates

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Workbook: Polar Coordinates WB7 Find a Polar equivalent for each of the following Cartesian equations:

Sketch the Cartesian graph (x and y axes)

a) y2=4 x

b) x2− y2=5

WB8 Find a Polar equivalent for the following Cartesian equation:

y √3=x+4

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Workbook: Polar Coordinates WB9 Find a Polar equivalent for each of the following Cartesian equations:

a) y=x2

b) ( x2+ y2 )2=4 xy

c) x cos∝+ y sin∝=p , p>0

d) y √3=x+4

e) y=1x

, x>0

f) x2+ y2=4

g) (x−1)2+( y−1)2=2

h)1x+ 1

y=1

a,a , x , y>0

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Workbook: Polar Coordinates

WB10 Some general equations

CIRCLES

r=a

r=acosθ

r=a sin θ

LINES

r=asec θ

r=acosec θ

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Workbook: Polar Coordinates WB11

Find the Cartesian equations of these polar functionsr=sin nθ and r=cosnθ

Polar Cartesian

r=sin θ

r=cosθ

r=sin 2θ

r=cos2θ

r=sin 3θ

r=cos3θ

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Workbook: Polar Coordinates WB12 Sketch these graphs with these polar equations, −π ≤θ ≤ π

a¿ r=2+cosθ CARDIOID

θ −56

π −23

π −12

π −13

π −16

π 0

r

θ 16

π 13

π 12

π 23

π 56

π π

r

b¿ r ¿2=16 cos 2θ LEMINSCATE

θ −56

π −23

π −12

π −13

π −16

π 0

r

θ 16

π 13

π 12

π 23

π 56

π π

r

c ¿ r=2eθ7 SPIRAL

θ −56

π −23

π −12

π −13

π −16

π 0

r

θ 16

π 13

π 12

π 23

π 56

π π

r

d ¿ r=4 sin 3θ ROSE

θ −56

π −23

π −12

π −13

π −16

π 0

r

θ 16

π 13

π 12

π 23

π 56

π π

r

WB13 Sketch these graphs with these polar equations, −π ≤θ ≤ π

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Workbook: Polar Coordinates

a¿ r=2 cosθ

b¿ r=2sin θ

c ¿ r=2−cos(θ−14

π )

d ¿r=2−sin θ

e ¿ r=2−sin2 θ

f ¿ r=2−sin 3 θ

WB14 Sketch the curve given by Polar equation:

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Workbook: Polar Coordinates

a) r=a

b) θ=a

c) r=aθ

d) r=a (1+cosθ )

e) r=asec θ

WB14 Sketch the curve given by Polar equation:

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Workbook: Polar Coordinates

f) r=sin 3θ

g) r2=a2 cos2θ

h) r=a (5+2cosθ )

i) r=a (3+2cosθ )

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Workbook: Polar Coordinates

WB15 Investigate

a) r=a( p+qcosθ)

b) r=cosnθ  ∧r=sin nθ  

c) r=tan nθ  ∧r=nθ

d) r=ncosθ  ∧r=n sinθ  

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