1. De Moivre’s Theorem

2. De Moivre’s Theorem
cos  i sin  n  cos n  i sin n
for all positive integers n

3. De Moivre’s Theorem
cos  i sin  n  cos n  i sin n
for all positive integers n
this extends to;

4. De Moivre’s Theorem
cos  i sin  n  cos n  i sin n
for all positive integers n
this extends to;
r cos  i sin  n  r n cos n  i sin n 

5. De Moivre’s Theorem
cos  i sin  n  cos n  i sin n
for all positive integers n
this extends to;
r cos  i sin  n  r n cos n  i sin n 
e.g .i  1  i 
5

6. De Moivre’s Theorem
cos  i sin  n  cos n  i sin n
for all positive integers n
this extends to;
r cos  i sin  n  r n cos n  i sin n 
e.g .i  1  i 
5
z  12   1
2
 2

7. De Moivre’s Theorem
cos  i sin  n  cos n  i sin n
for all positive integers n
this extends to;
r cos  i sin  n  r n cos n  i sin n 
e.g .i  1  i    1
5
z  12   1
2 1
arg z  tan  
 1 
 2


4

8. De Moivre’s Theorem
cos  i sin  n  cos n  i sin n
for all positive integers n
this extends to;
r cos  i sin  n  r n cos n  i sin n 
e.g .i  1  i    1
5
z  12   1
2 1
arg z  tan  
   
5
 1 
  2cis    2
  4  

4

9. De Moivre’s Theorem
cos  i sin  n  cos n  i sin n
for all positive integers n
this extends to;
r cos  i sin  n  r n cos n  i sin n 
e.g .i  1  i    1
5
z  12   1
2 1
arg z  tan  
   
5
 1 
  2cis    2
  4  

4
5
  2  cis  
5
 
 4 

10. De Moivre’s Theorem
cos  i sin  n  cos n  i sin n
for all positive integers n
this extends to;
r cos  i sin  n  r n cos n  i sin n 
e.g .i  1  i    1
5
z  12   1
2 1
arg z  tan  
   
5
 1 
  2cis    2
  4  

4
5
  2  cis  
5
 
 4 
 3 
 4 2cis 
 4 

11.  cos 3  i sin 3 
 4 2 
 4 4 

12.  cos 3  i sin 3 
 4 2 
 4 4 
 4 2 
1 1 
  i
 2 2 

13.  4 2 cos 3  i sin 3 

 4 4 
 4 2 
1 1 
  i
 2 2 
 4  4i

14.  4 2 cos 3  i sin 3 

 4 4 
 4 2 
1 1 
  i
 2 2 
 4  4i
ii  Express cos 2 and sin 2 in terms of cos and sin 

15.  4 2 cos 3  i sin 3 

 4 4 
 4 2 
1 1 
  i
 2 2 
 4  4i
ii  Express cos 2 and sin 2 in terms of cos and sin 
cos 2  i sin 2  cos  i sin  
2

16.  4 2 cos 3  i sin 3 

 4 4 
 4 2 
1 1 
  i
 2 2 
 4  4i
ii  Express cos 2 and sin 2 in terms of cos and sin 
cos 2  i sin 2  cos  i sin  
2
 cos 2   2i sin  cos  sin 2 

17.  4 2 cos 3  i sin 3 

 4 4 
 4 2 
1 1 
  i
 2 2 
 4  4i
ii  Express cos 2 and sin 2 in terms of cos and sin 
cos 2  i sin 2  cos  i sin  
2
 cos 2   2i sin  cos  sin 2 
By equating real and imaginary parts

18.  4 2 cos 3  i sin 3 

 4 4 
 4 2 
1 1 
  i
 2 2 
 4  4i
ii  Express cos 2 and sin 2 in terms of cos and sin 
cos 2  i sin 2  cos  i sin  
2
 cos 2   2i sin  cos  sin 2 
By equating real and imaginary parts
cos 2  cos 2   sin 2 

19.  4 2 cos 3  i sin 3 

 4 4 
 4 2 
1 1 
  i
 2 2 
 4  4i
ii  Express cos 2 and sin 2 in terms of cos and sin 
cos 2  i sin 2  cos  i sin  
2
 cos 2   2i sin  cos  sin 2 
By equating real and imaginary parts
cos 2  cos 2   sin 2  sin 2  2 sin  cos

20. Finding Roots

21. Finding Roots
If z n  x  iy
z n  rcis

22. Finding Roots
If z n  x  iy
z n  rcis
 2k   
z  n rcis  k  0,1,, n  1
 n  

23. Finding Roots
If z n  x  iy
z n  rcis
 2k   
z  n rcis  k  0,1,, n  1
 n  
e.g .i  z 2  4i

24. Finding Roots
If z n  x  iy
z n  rcis
2k   
z  n rcis  k  0,1,, n  1
 n 
 
e.g .i  z 2  4i

z 2  4cis
2

25. Finding Roots
If z n  x  iy
z n  rcis
2k   
z  n rcis  k  0,1,, n  1
 n 
 
e.g .i  z 2  4i

z 2  4cis
2
 2k   
 2
z  2cis  k  0,1
2 
 
 

26. Finding Roots
If z n  x  iy
z n  rcis
2k   
z  n rcis  k  0,1,, n  1
 n 
 
e.g .i  z 2  4i

z 2  4cis
2
 2k   
 2
z  2cis  k  0,1
2 
 
 
 5
z  2cis ,2cis
4 4

27. Finding Roots
If z n  x  iy
z n  rcis
2k   
z  n rcis  k  0,1,, n  1
 n 
 
e.g .i  z 2  4i

z  2
1 1   1 1 
z 2  4cis   i ,2   i
2  2 2   2 2 
 2k   
 2
z  2cis  k  0,1
2 
 
 
 5
z  2cis ,2cis
4 4

28. Finding Roots
If z n  x  iy
z n  rcis
2k   
z  n rcis  k  0,1,, n  1
 n 
 
e.g .i  z 2  4i

z  2
1 1   1 1 
z 2  4cis   i ,2   i
2  2 2   2 2 
 2k    z  2  2i, 2  2i
 2
z  2cis  k  0,1
2 
 
 
 5
z  2cis ,2cis
4 4

29. ii  x 4  8  0

30. ii  x 4  8  0
x4  8

31. ii  x 4  8  0
x4  8
x 4  8cis 0

32. ii  x 4  8  0
x4  8
x 4  8cis 0
4  2k 
x  8cis  k  0,1,2,3
 4  

33. ii  x 4  8  0
x4  8
x 4  8cis 0
4  2k 
x  8cis  k  0,1,2,3
 4  
 3
x  8cis 0, 8cis , 8cis , 8cis
4 4 4 4
2 2

34. ii  x 4  8  0
x4  8
x 4  8cis 0
4  2k 
x  8cis  k  0,1,2,3
 4  
 3
x  8cis 0, 8cis , 8cis , 8cis
4 4 4 4
2 2
x  4 8, 4 8i, 4 8, 4 8i

35. ii  x 4  8  0
x4  8
x 4  8cis 0
4  2k 
x  8cis  k  0,1,2,3
 4  
 3
x  8cis 0, 8cis , 8cis , 8cis
4 4 4 4
2 2
x  4 8, 4 8i, 4 8, 4 8i
1 1
iii  If z  cos  i sin  , find z  n and z  n
n n
z z

36. ii  x 4  8  0
x4  8
x 4  8cis 0
4  2k 
x  8cis  k  0,1,2,3
 4  
 3
x  8cis 0, 8cis , 8cis , 8cis
4 4 4 4
2 2
x  4 8, 4 8i, 4 8, 4 8i
1 1
iii  If z  cos  i sin  , find z  n and z  n
n n
z z
z  cos n  i sin n
n

37. ii  x 4  8  0
x4  8
x 4  8cis 0
4  2k 
x  8cis  k  0,1,2,3
 4  
 3
x  8cis 0, 8cis , 8cis , 8cis
4 4 4 4
2 2
x  4 8, 4 8i, 4 8, 4 8i
1 1
iii  If z  cos  i sin  , find z  n and z  n
n n
z z
z  cos n  i sin n
n
1
 z n  cos n   i sin  n 
zn

38. ii  x 4  8  0
x4  8
x 4  8cis 0
4  2k 
x  8cis  k  0,1,2,3
 4  
 3
x  8cis 0, 8cis , 8cis , 8cis
4 4 4 4
2 2
x  4 8, 4 8i, 4 8, 4 8i
1 1
iii  If z  cos  i sin  , find z  n and z  n
n n
z z
z  cos n  i sin n
n
1
 z n  cos n   i sin  n 
zn
 cos n  i sin n

39. ii  x 4  8  0
x4  8
x 4  8cis 0
4  2k 
x  8cis  k  0,1,2,3
 4  
 3
x  8cis 0, 8cis , 8cis , 8cis
4 4 4 4
2 2
x  4 8, 4 8i, 4 8, 4 8i
1 1
iii  If z  cos  i sin  , find z  n and z  n
n n
z z
z  cos n  i sin n
n
1
 z n  cos n   i sin  n 
zn
 cos n  i sin n cos is even function  cos x   cos x 
 
sin is odd function  sin  x    sin x 

40. ii  x 4  8  0
x4  8
x 4  8cis 0
4  2k 
x  8cis  k  0,1,2,3
 4  
 3
x  8cis 0, 8cis , 8cis , 8cis
4 4 4 4
2 2
x  4 8, 4 8i, 4 8, 4 8i
1 1
iii  If z  cos  i sin  , find z  n and z  n
n n
z z
z  cos n  i sin n
n
1
 z n  cos n   i sin  n 
zn
 cos n  i sin n cos is even function  cos x   cos x 
 
sin is odd function  sin  x    sin x 
1
z  n  2 cos n
n
z

41. ii  x 4  8  0
x4  8
x 4  8cis 0
4  2k 
x  8cis  k  0,1,2,3
 4  
 3
x  8cis 0, 8cis , 8cis , 8cis
4 4 4 4
2 2
x  4 8, 4 8i, 4 8, 4 8i
1 1
iii  If z  cos  i sin  , find z  n and z  n
n n
z z
z  cos n  i sin n
n
1
 z n  cos n   i sin  n 
zn
 cos n  i sin n cos is even function  cos x   cos x 
 
sin is odd function  sin  x    sin x 
1 1
z  n  2 cos n
n
z  n  2i sin n
n
z z

42. iv  Express cos3  in terms of cos n

43. iv  Express cos3  in terms of cos n
3
2 cos 3  1
z 
 z

44. iv  Express cos3  in terms of cos n
3
2 cos 3  1
z 
 z
3 1
8 cos   z  3 z   3
3 3
z z

45. iv  Express cos3  in terms of cos n
3
2 cos 3  1
z 
 z
3 1
8 cos   z  3 z   3
3 3
z z
 z 3  1   3 z  1 
  
 z3   z

46. iv  Express cos3  in terms of cos n
3
2 cos 3  1
z 
 z
3 1
8 cos   z  3 z   3
3 3
z z
 z 3  1   3 z  1 
  
 z3   z
 2 cos 3  6 cos

47. iv  Express cos3  in terms of cos n
3
2 cos 3  1
z 
 z
3 1
8 cos   z  3 z   3
3 3
z z
 z 3  1   3 z  1 
  
 z3   z
 2 cos 3  6 cos
1 3
 cos3   cos 3  cos
4 4

48. Alternative for finding x  iy

49. Alternative for finding x  iy
If x  iy  a  ib

50. Alternative for finding x  iy
If x  iy  a  ib
x  x  iy
then a 
2

51. Alternative for finding x  iy
If x  iy  a  ib
x  x  iy y
then a  b
2 2a

52. Alternative for finding x  iy
If x  iy  a  ib
x  x  iy y
then a  b
2 2a
e.g . Find  12  16i

53. Alternative for finding x  iy
If x  iy  a  ib
x  x  iy y
then a  b
2 2a
e.g . Find  12  16i
 12  16i  12 2  16 2
 20

54. Alternative for finding x  iy
If x  iy  a  ib
x  x  iy y
then a  b
2 2a
e.g . Find  12  16i
 12  16i  12 2  16 2
 20
 12  20
a
2
8

2
2

55. Alternative for finding x  iy
If x  iy  a  ib
x  x  iy y
then a  b
2 2a
e.g . Find  12  16i
 12  16i  12 2  16 2
 20
 12  20 16
a b
2 4
8 4

2
2

56. Alternative for finding x  iy
If x  iy  a  ib
x  x  iy y
then a  b
2 2a
e.g . Find  12  16i
 12  16i  12 2  16 2
 20
 12  20 16
a b
2 4
8 4

2
  12  16i  2  4i 
2

57. Exercise 4D; evens
Exercise 4E; 1 to 4 ac
Exercise 4F; 1 to 4, 5ab, 10