Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
This will help you on how to solve quadratic equations by factoring.
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This will help you on how to solve quadratic equations by factoring.
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Power Series,Taylor's and Maclaurin's SeriesShubham Sharma
A details explanation about Taylor's and Maclaurin's series with variety of examples are included in this slide. The aim is to give the viewer the basic knowledge about the topic.
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. 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. 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. 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. 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. 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. 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. 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. Finding Roots
If z n x iy
z n rcis
2k
z rcis
n
n
k 0,1,, n 1
11. Finding Roots
If z n x iy
z n rcis
2k
z rcis
n
n
e.g .i z 2 4i
k 0,1,, n 1
12. Finding Roots
If z n x iy
z n rcis
2k
z rcis
n
n
e.g .i z 2 4i
2
z 4cis
2
k 0,1,, n 1
13. Finding Roots
If z n x iy
z n rcis
2k
z rcis
n
n
e.g .i z 2 4i
2
z 4cis
2
2k
2
z 2cis
2
k 0,1
k 0,1,, n 1
14. Finding Roots
If z n x iy
z n rcis
2k
z rcis
n
n
e.g .i z 2 4i
2
z 4cis
2
2k
2
z 2cis
2
5
z 2cis ,2cis
4
4
k 0,1
k 0,1,, n 1
15. Finding Roots
If z n x iy
z n rcis
2k
z rcis
n
n
e.g .i z 2 4i
2
z 4cis
2
2k
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. Finding Roots
If z n x iy
z n rcis
2k
z rcis
n
n
e.g .i z 2 4i
OR
2
z 4cis
2
2k
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. Finding Roots
If z n x iy
z n rcis
2k
z rcis
n
n
e.g .i z 2 4i
OR
2
z 4cis
2
2k
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. Finding Roots
If z n x iy
z n rcis
2k
z rcis
n
n
k 0,1,, n 1
e.g .i z 2 4i
OR
2
y
z 4cis
2
2cis
4
2k
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
21. ii
x 4 16 0
x 4 16
x 4 16cis 0
2 k
x 2cis
4
k 0,1, 2,3
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. 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. 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. 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. 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. 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