Upcoming SlideShare
×

# X2 t01 09 de moivres theorem

1,666 views

Published on

1 Like
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

Views
Total views
1,666
On SlideShare
0
From Embeds
0
Number of Embeds
229
Actions
Shares
0
83
0
Likes
1
Embeds 0
No embeds

No notes for slide

### X2 t01 09 de moivres theorem

1. 1. De Moivre’s Theorem  cos  i sin    cos n  i sin n n for all integers n
2. 2. De Moivre’s Theorem  cos  i sin    cos n  i sin n n for all integers n this extends to; r cos  i sin   n  r n cos n  i sin n 
3. 3. De Moivre’s Theorem  cos  i sin    cos n  i sin n n for all integers n this extends to; r cos  i sin   n e.g . 1  i  5  r n cos n  i sin n 
4. 4. De Moivre’s Theorem  cos  i sin    cos n  i sin n n for all integers n this extends to; r cos  i sin   n e.g . 1  i  5  r n cos n  i sin n  z  12   1 2  2   1 arg z  tan    1  1   4
5. 5. De Moivre’s Theorem  cos  i sin    cos n  i sin n n for all integers n this extends to; r cos  i sin   n e.g . 1  i  5 z  12   1 2        2cis    r n cos n  i sin n   4  5  2   1 arg z  tan    1  1   4
6. 6. De Moivre’s Theorem  cos  i sin    cos n  i sin n n for all integers n this extends to; r cos  i sin   n e.g . 1  i  5  z  12   1 2        2cis    r n cos n  i sin n   4  5  2   1 arg z  tan   5  1    5  2  cis    4   4 1
7. 7. De Moivre’s Theorem  cos  i sin    cos n  i sin n n for all integers n this extends to; r cos  i sin   n e.g . 1  i  5 z  12   1 2        2cis    r n cos n  i sin n   4  5  2   1 arg z  tan   5  1    5    2  cis    4   4 3   4 2cis    4  1
8. 8. De Moivre’s Theorem  cos  i sin    cos n  i sin n n for all integers n this extends to; r cos  i sin   n e.g . 1  i  5 z  12   1 2        2cis    r n cos n  i sin n   4  5  2   1 arg z  tan   5  1    5    2  cis    4   4 3   4 2cis    4  1 1  i  5  cos 3  i sin 3   4 2  4 4  
9. 9. De Moivre’s Theorem  cos  i sin    cos n  i sin n n for all integers n this extends to; r cos  i sin   n e.g . 1  i  5 z  12   1 2        2cis    r n cos n  i sin n   4  5  2   1 arg z  tan   5  1    5    2  cis    4   4 3   4 2cis    4  1 1  i  5  cos 3  i sin 3   4 2  4 4   1 1   4 2   i  2 2    4  4i
10. 10. Finding Roots If z n  x  iy z n  rcis  2k    z  rcis   n   n k  0,1,, n  1
11. 11. Finding Roots If z n  x  iy z n  rcis  2k    z  rcis   n   n e.g .i  z 2  4i k  0,1,, n  1
12. 12. Finding Roots If z n  x  iy z n  rcis  2k    z  rcis    n  n e.g .i  z 2  4i  2 z  4cis 2 k  0,1,, n  1
13. 13. Finding Roots If z n  x  iy z n  rcis  2k    z  rcis    n  n e.g .i  z 2  4i  2 z  4cis 2  2k     2 z  2cis  2      k  0,1 k  0,1,, n  1
14. 14. Finding Roots If z n  x  iy z n  rcis  2k    z  rcis    n  n e.g .i  z 2  4i  2 z  4cis 2  2k     2 z  2cis  2      5  z  2cis ,2cis 4 4 k  0,1 k  0,1,, n  1
15. 15. Finding Roots If z n  x  iy z n  rcis  2k    z  rcis    n  n e.g .i  z 2  4i  2 z  4cis 2  2k     2  k  0,1 z  2cis  2      5  z  2cis ,2cis 4 4  1  1 i ,2  1  1 i  z  2    2   2 2   2 k  0,1,, n  1 z  2  2i, 2  2i
16. 16. Finding Roots If z n  x  iy z n  rcis  2k    z  rcis    n  n e.g .i  z 2  4i  OR 2 z  4cis 2  2k     2  k  0,1 z  2cis  2      5  z  2cis ,2cis 4 4  1  1 i ,2  1  1 i  z  2    2   2 2   2 k  0,1,, n  1 y x z  2  2i, 2  2i
17. 17. Finding Roots If z n  x  iy z n  rcis  2k    z  rcis    n  n e.g .i  z 2  4i  OR 2 z  4cis 2  2k     2  k  0,1 z  2cis  2      5  z  2cis ,2cis 4 4  1  1 i ,2  1  1 i  z  2    2   2 2   2 k  0,1,, n  1 y 2cis  x z  2  2i, 2  2i 4
18. 18. Finding Roots If z n  x  iy z n  rcis  2k    z  rcis    n  n k  0,1,, n  1 e.g .i  z 2  4i  OR 2 y z  4cis  2 2cis 4  2k     2  k  0,1 z  2cis  2  x     3 2cis  5  z  2cis ,2cis 4 4 4  1  1 i ,2  1  1 i  z  2 z  2  2i, 2  2i    2   2 2   2
19. 19.  ii  x 4  16  0
20. 20.  ii  x 4  16  0 x 4  16 x 4  16cis 0
21. 21.  ii  x 4  16  0 x 4  16 x 4  16cis 0  2 k  x  2cis   4   k  0,1, 2,3
22. 22.  ii  x 4  16  0 x 4  16 x 4  16cis 0  2 k  x  2cis   4    k  0,1, 2,3 3 x  2cis 0, 2cis , 2cis , 2cis 2 2
23. 23.  ii  x 4  16  0 x 4  16 x 4  16cis 0  2 k  x  2cis   4   k  0,1, 2,3  3 x  2cis 0, 2cis , 2cis , 2cis 2 2 x  2, 2i, 2, 2i
24. 24.  ii  x 4  16  0 x 4  16 x 4  16cis 0  2 k  x  2cis   4   k  0,1, 2,3  3 x  2cis 0, 2cis , 2cis , 2cis 2 2 x  2, 2i, 2, 2i OR y x
25. 25.  ii  x 4  16  0 x 4  16 x 4  16cis 0  2 k  x  2cis   4   k  0,1, 2,3  3 x  2cis 0, 2cis , 2cis , 2cis 2 2 x  2, 2i, 2, 2i OR y 2cis 0 x
26. 26.  ii  x 4  16  0 x 4  16 x 4  16cis 0  2 k  x  2cis   4   k  0,1, 2,3  3 x  2cis 0, 2cis , 2cis , 2cis 2 2 x  2, 2i, 2, 2i OR y 2cis  2 2cis 2cis 0 x 2cis   2
27. 27.  ii  x 4  16  0 x 4  16 x 4  16cis 0  2 k  x  2cis   4   k  0,1, 2,3  3 x  2cis 0, 2cis , 2cis , 2cis 2 2 x  2, 2i, 2, 2i OR y 2cis Patel: Exercise 4E; 1 to 4 ac  2 2cis 2cis 0 x 2cis  Cambridge: Exercise 7A; 1, 2, 3 abef, 5, 6, 7, 9 to 14, 16 to 18  2