This document provides information about polar coordinates including: - Relations between Cartesian and polar coordinates - Sketching graphs in polar coordinates such as circles, cardioids, and roses - Finding intersections of curves, slopes of tangents, and areas bounded by polar curves - Computing arc lengths and surfaces of revolution generated by polar curves It discusses key concepts like symmetry properties and provides examples of computing specific values related to polar curves.

1. Polar Coordinates
Advanced Level Pure Mathematics
TARUNGEHLOT
Introduction 2
Relations between Cartesian and Polar Coordinates 3
Sketch of Graphs in Polar Coordinates 5
Intersection of Two Curves in Polar Coordinates 10
Slope of a Tangent 11
Areas 13
Arc Lengths 15
Page 1

2. Polar Coordinates
Advanced Level Pure Mathematics
Surfaces of Revolution 16
Introduction
Cartesian Coordinate plane Polar Coordinate plane
y
p (r,θ)
Page 2

3. Polar Coordinates
Advanced Level Pure Mathematics
p (x, y)
y vectorial angle θ
x x 0 origin polor axis
(pole) (initial axis)
Example
y
P( 2 , 45 )
p (1, 1) 2
1 x 0 45
1
N.B. (1) is positive if it is measured anti-clockwise.
(2) is negative if it is measured clockwise.
0
P (4, ) -
6 6
4
Q (4, - )
6 6
0
(3) r can be negative.
A( 2 , ) A( 2 , )
3 3
Page 3

4. Polar Coordinates
Advanced Level Pure Mathematics
2 2
3
2
3
0 B (- 2 , )
3
(4) If r > 0 , P ( r, ) = P ( r, 2n + ) , n Z.
e.g. P ( r, ) = P ( r, 3 ) = P ( r, - )
p (r, 3 )
p (r, )
r
p (r, - ) 0
(5) If r = 0 , O ( 0, ) is called the pole or origin , R.
(6) If r > 0 , Q ( -r, ) = Q ( r, + ) .
3
0
4
Q (4, + ) or Q (-4, )
3 3
Page 4

5. Polar Coordinates
Advanced Level Pure Mathematics
(7) Q is an imaginary point (which r<0) and P is a real point (which r>0) .
Relations between Cartesian and Polar Coordinates
2 2
r x y
x r co s
y y
y r sin tan
1
x
P (x, y)  P (r, )
r y
x (polar axis)
0 x
Example Express the Cartesian coordinates in polar coordinates
(a) ( 2 3 , 2) , (b) ( 2 , 2)
2 2
(a) r x y 12 4 4
1 2 1 1
tan tan
2 3 3 6
so the polar coordinates= ( 4 , )
6
(b)
Page 5

6. Polar Coordinates
Advanced Level Pure Mathematics
Example Express the polar coordinates in Cartesian coordinates
(a) (-2 , ) , (b) (2 3 , )
6
Page 6

7. Polar Coordinates
Advanced Level Pure Mathematics
Definition Polar Equations : r = f( )
Example r=2cos , r=2(1-sin ) , r=3 , etc.
Example Investigate the loci represented by the following polar equations.
(a) = , where is a constant.
(b) r = a , where a is a constant.
1 y
(a) = tan
x
y
tan m (say)
x
y mx
it is a straight line with inclination to the initial line and passing through the pole.
(b)
Page 7

8. Polar Coordinates
Advanced Level Pure Mathematics
Example Express the following loci in polar coordinates :
(a) The straight line with normal form x cos y sin p.
2 2
(b) The circle x y 2 ax 0.
Page 8

9. Polar Coordinates
Advanced Level Pure Mathematics
Sketch of Graphs in Polar Coordinates
Example Plot the polar curve r 2 cos for 0 2 .
Solution
3 3
0 …. …. 2
4 3 2 4 2
r 2 2 1 0 - 2 -2 …. 0 …. 2
y
P( r, )
r
0 x (polar axis)
1 2
r=2cos
2 2 x
N.B. (1) r 2 cos x y 2
2 2
x y
2 2 2
(x 1) (y 0) 1 (which is a circle)
Page 9

10. Polar Coordinates
Advanced Level Pure Mathematics
(2) If f ( ) f ( ) , the polar curve is symmetric with respect to the initial line.
(r, )
0 e.g. r=2cos , r=2(1-cos )
-
(r ,- )
(3) If f ( ) f ( ) , or f ( ) f( ) , the curve is symmetric with respect to the vertical line .
2
(-r, - )=(r, - ) (r, )
e.g. r=2sin , r=a(1+sin )
-
0
(4) If f ( ) f( ) , or g (r ) g ( r ) , the curve is symmetric with respect to the pole.
(r, )
0 e.g. r2 = a2 sin2
Page 10

11. Polar Coordinates
Advanced Level Pure Mathematics
(-r, )
=(r, + )
I. Cardioids
Example Sketch the graph of the polar equation r 2 (1 cos ) for 0 2 .
Solution Let f( ) = 2(1-cos )
f( ) = 2[1-cos(- )] = f( )
The graph is symmetric wiyh respect to the initial line.
0 2 3 5
6 4 3 2 3 4 6
r 0 2- 3 2- 2 1 2 3 2+ 2 2+ 3 4
The graph is
R=2(1-cos )
(4, ) 0
Example Sketch the graph of the polar equation r 2 (1 cos ) for 0 2 .
Page 11

12. Polar Coordinates
Advanced Level Pure Mathematics
Since f(- )=2[1+cos(- )]=2(1+cos )=f( ) so the graph is symmetric w.r.t. the initial line.
N.B. r=a(1+sin ) r=a(1-sin )
(2a, ) (0, ) 0 (a, 0)
2
(a, ) 0 (a, 0)
3
(2a , )
2
Page 12

13. Polar Coordinates
Advanced Level Pure Mathematics
II. Limacons
Example Sketch the graph of the polar equation r 2 4 cos .
N.B. r=a+bcos (0<a<b) r=a+bcos (0<b<a)
0 0
Page 13

14. Polar Coordinates
Advanced Level Pure Mathematics
r=a+bsin (0<b<a) r=a+bsin (0<a<b)
0
0
III. n-Leafed Roses
r=acos3 (a>0 , 0 2 ) r=asin3 (a>0 , 0 2 )
2
= =
3 3
=
6
5
=
6
r=acos2 (a>0 , 0 2 ) r=asin2 (a>0 , 0 2 )
Page 14

15. Polar Coordinates
Advanced Level Pure Mathematics
Example Sketch the graph of the polar equation r a sin 3 , where a>0.
Example Sketch the graph of the polar equation r a cos 2 , where a>0.
Page 15

16. Polar Coordinates
Advanced Level Pure Mathematics
IV. Two-Leafed Lemniscates
r2=a2cos2 (a>0 , 0 2 ) r2=a2sin2 (a>0 ,
0 2 )
V. Spirals of Archimedes.
a
r=a (a>0 , 0) r e (a>0 , 0)
Page 16

17. Polar Coordinates
Advanced Level Pure Mathematics
a
r e (a<0 , 0)
Intersection of Two Curves in Polar Coordinates
Example Find the points of intersection of the circle r 2 and the cardioid r 2 (1 cos ) .
Page 17

18. Polar Coordinates
Advanced Level Pure Mathematics
Example Find the points of intersection of the circle r 2 cos and the four-leafed rose r sin 2 .
Page 18

19. Polar Coordinates
Advanced Level Pure Mathematics
Slope of a Tangent
dx dr
cos r sin
x r cos d d
y r sin dy dr
sin r cos
d d
dr
sin r cos
dy d
dx dr
cos r sin
d
dr
tan r
d
dr
r tan
d
The angle between the tangent at P to the polar
Page 19

20. Polar Coordinates
Advanced Level Pure Mathematics
curve r=f( ) and radius vector OP is given by
y r=f( )
r
tan tangent at point p
dr
d
p(r, )
x
0
Page 20

21. Polar Coordinates
Advanced Level Pure Mathematics
Example Given the polar curve r 2 2 cos , 0 2 .
(a) Find (i) the slope of tangent at ,
4
(ii) the points at which the tangent is horizontal,
(iii) the points at which the tangent is vertical,
(iv) the values of at which the angle between the radius vector and tangent is .
4
(b) Sketch the polar curve.
Page 21

22. Polar Coordinates
Advanced Level Pure Mathematics
Page 22

23. Polar Coordinates
Advanced Level Pure Mathematics
Areas
Theorem The area enclosed by the polar curve r f ( ) betweem the lines = and = is given by
r=f( ) =
1 2
A r d =
2
where r f( ) 0 on [ , ].
0
Example Find the area enclosed by the cardioid r a (1 cos ) , a>0.
Page 23

24. Polar Coordinates
Advanced Level Pure Mathematics
Example Find the area of the inner loop of the curve r 2 4 cos .
2 4
Solution Since the inner loop bounded by the curve corresponds to the interval , .
3 3
N.B. (1) r f ( ), r g( )
1 2 2 1 2 2
Area enclosed = f( ) g( ) d = f( ) g( ) d
2 2
=
Page 24

25. Polar Coordinates
Advanced Level Pure Mathematics
=
r = f( )
r = g( )
(2) If the pole O lies within a closed polar curve, then
2 1 2
the area enclosed= r d .
0 2
Example (a) Find the area inside the circle r 5 cos and outside the limacon r 2 cos .
(b) Find the area inside the circle r 5 cos and the limacon r 2 cos .
Page 25

26. Polar Coordinates
Advanced Level Pure Mathematics
Arc Lengths
The arc length S of the curve r f ( ) form = to = is
S = ds dy ds
2 2
= dx dy dx
Page 26

27. Polar Coordinates
Advanced Level Pure Mathematics
2 2
dx dy
= d r=f( )
d d
2 2
dr dr
= cos r sin sin r cos d
d d
2
dr 2
= r d
d
provided that r is differentiable on ( , ) and the curve does not intersect itself on ( , ).
Example Find the length of the circumference of the cardioid r a (1 cos ) , where a>0.
Page 27

28. Polar Coordinates
Advanced Level Pure Mathematics
Example Find the length of the curve of the circle r 5 cos that is outside the limacon r 2 cos .
Page 28

29. Polar Coordinates
Advanced Level Pure Mathematics
Surfaces of Revolution
Theorem The area of surface revolution S of the curve from = to = about the inital line is
S 2 yds
2
dr 2
2 r sin r d r=f( )
d
ds
y
0
Theorem The area of surface revolution S of the curve from = to = about the line (i.e. y-axis) is
2
S 2 xds
2
dr 2
2 r cos r d
d
Page 29

30. Polar Coordinates
Advanced Level Pure Mathematics
x ds
0
Example Find the surface area generated by revolving the circle r 2a sin , a>0 about the line .
2
Page 30

31. Polar Coordinates
Advanced Level Pure Mathematics
Example Find the surface area generated by revolving the cardioid r a (1 cos ) , where a>0 , about the initial line.
Page 31

32. Polar Coordinates
Advanced Level Pure Mathematics
Example The equation of a curve C in polar coordinates is
r 1 sin , 0 2 .
(a) Sketch curve C.
(b) Find the area bounded by curve C. [HKAL94] (5 marks)
Page 32

33. Polar Coordinates
Advanced Level Pure Mathematics
Example
y
Page 33

34. Polar Coordinates
Advanced Level Pure Mathematics
p
Q
0 a 2a x
Let C be the circle given by the polar equation r=2acos (where a>0) , P be a variable point on C and O be the origin. Let Q be a point lying on the line through O and P such
that P and Q are on the same side of O and
2
OP OQ a .
a
Show that the Cartesian equation of the locus of Q is x .
2
Page 34