x2 t01 09 de moivres theorem
TRANSCRIPT
![Page 1: X2 t01 09 de moivres theorem](https://reader033.vdocument.in/reader033/viewer/2022060116/557e77cdd8b42a4d108b4b69/html5/thumbnails/1.jpg)
De Moivre’s Theorem cos sin cos sin
for all integers
ni n i nn
![Page 2: X2 t01 09 de moivres theorem](https://reader033.vdocument.in/reader033/viewer/2022060116/557e77cdd8b42a4d108b4b69/html5/thumbnails/2.jpg)
De Moivre’s Theorem cos sin cos sin
for all integers
ni n i nn
ninrir nn sincossincos
this extends to;
![Page 3: X2 t01 09 de moivres theorem](https://reader033.vdocument.in/reader033/viewer/2022060116/557e77cdd8b42a4d108b4b69/html5/thumbnails/3.jpg)
De Moivre’s Theorem cos sin cos sin
for all integers
ni n i nn
ninrir nn sincossincos
this extends to;
5. . 1e g i
![Page 4: X2 t01 09 de moivres theorem](https://reader033.vdocument.in/reader033/viewer/2022060116/557e77cdd8b42a4d108b4b69/html5/thumbnails/4.jpg)
De Moivre’s Theorem cos sin cos sin
for all integers
ni n i nn
ninrir nn sincossincos
this extends to;
5. . 1e g i 2
11 22
z
4
11tanarg 1
z
![Page 5: X2 t01 09 de moivres theorem](https://reader033.vdocument.in/reader033/viewer/2022060116/557e77cdd8b42a4d108b4b69/html5/thumbnails/5.jpg)
De Moivre’s Theorem cos sin cos sin
for all integers
ni n i nn
ninrir nn sincossincos
5
42
cis
this extends to;
5. . 1e g i 2
11 22
z
4
11tanarg 1
z
![Page 6: X2 t01 09 de moivres theorem](https://reader033.vdocument.in/reader033/viewer/2022060116/557e77cdd8b42a4d108b4b69/html5/thumbnails/6.jpg)
De Moivre’s Theorem cos sin cos sin
for all integers
ni n i nn
ninrir nn sincossincos
5
42
cis
this extends to;
5. . 1e g i 2
11 22
z
452 5 cis
4
11tanarg 1
z
![Page 7: X2 t01 09 de moivres theorem](https://reader033.vdocument.in/reader033/viewer/2022060116/557e77cdd8b42a4d108b4b69/html5/thumbnails/7.jpg)
De Moivre’s Theorem cos sin cos sin
for all integers
ni n i nn
ninrir nn sincossincos
5
42
cis
this extends to;
5. . 1e g i 2
11 22
z
452 5 cis
4
11tanarg 1
z
4324 cis
![Page 8: X2 t01 09 de moivres theorem](https://reader033.vdocument.in/reader033/viewer/2022060116/557e77cdd8b42a4d108b4b69/html5/thumbnails/8.jpg)
De Moivre’s Theorem cos sin cos sin
for all integers
ni n i nn
ninrir nn sincossincos
5
42
cis
this extends to;
5. . 1e g i 2
11 22
z
452 5 cis
4
11tanarg 1
z
4324 cis
43sin
43cos24 i
51 i
![Page 9: X2 t01 09 de moivres theorem](https://reader033.vdocument.in/reader033/viewer/2022060116/557e77cdd8b42a4d108b4b69/html5/thumbnails/9.jpg)
De Moivre’s Theorem cos sin cos sin
for all integers
ni n i nn
ninrir nn sincossincos
5
42
cis
this extends to;
5. . 1e g i 2
11 22
z
452 5 cis
4
11tanarg 1
z
4324 cis
43sin
43cos24 i
i
21
2124
i44
51 i
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Finding Rootsrcisziyxz
n
n
If
1,,1,0 2
nknkcisrz n
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Finding Rootsrcisziyxz
n
n
If
1,,1,0 2
nknkcisrz n
izige 4 .. 2
![Page 12: X2 t01 09 de moivres theorem](https://reader033.vdocument.in/reader033/viewer/2022060116/557e77cdd8b42a4d108b4b69/html5/thumbnails/12.jpg)
Finding Rootsrcisziyxz
n
n
If
1,,1,0 2
nknkcisrz n
izige 4 .. 2
242 cisz
![Page 13: X2 t01 09 de moivres theorem](https://reader033.vdocument.in/reader033/viewer/2022060116/557e77cdd8b42a4d108b4b69/html5/thumbnails/13.jpg)
Finding Rootsrcisziyxz
n
n
If
1,,1,0 2
nknkcisrz n
izige 4 .. 2
242 cisz
1,0 2
22
2
k
kcisz
![Page 14: X2 t01 09 de moivres theorem](https://reader033.vdocument.in/reader033/viewer/2022060116/557e77cdd8b42a4d108b4b69/html5/thumbnails/14.jpg)
Finding Rootsrcisziyxz
n
n
If
1,,1,0 2
nknkcisrz n
izige 4 .. 2
242 cisz
1,0 2
22
2
k
kcisz
452,
42 ciscisz
![Page 15: X2 t01 09 de moivres theorem](https://reader033.vdocument.in/reader033/viewer/2022060116/557e77cdd8b42a4d108b4b69/html5/thumbnails/15.jpg)
Finding Rootsrcisziyxz
n
n
If
1,,1,0 2
nknkcisrz n
izige 4 .. 2
242 cisz
1,0 2
22
2
k
kcisz
452,
42 ciscisz
iiz
21
212,
21
212 iiz 22,22
![Page 16: X2 t01 09 de moivres theorem](https://reader033.vdocument.in/reader033/viewer/2022060116/557e77cdd8b42a4d108b4b69/html5/thumbnails/16.jpg)
Finding Rootsrcisziyxz
n
n
If
1,,1,0 2
nknkcisrz n
izige 4 .. 2
242 cisz
1,0 2
22
2
k
kcisz
452,
42 ciscisz
iiz
21
212,
21
212 iiz 22,22
y
x
OR
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Finding Rootsrcisziyxz
n
n
If
1,,1,0 2
nknkcisrz n
izige 4 .. 2
242 cisz
1,0 2
22
2
k
kcisz
452,
42 ciscisz
iiz
21
212,
21
212 iiz 22,22
y
x
OR2
4cis
![Page 18: X2 t01 09 de moivres theorem](https://reader033.vdocument.in/reader033/viewer/2022060116/557e77cdd8b42a4d108b4b69/html5/thumbnails/18.jpg)
Finding Rootsrcisziyxz
n
n
If
1,,1,0 2
nknkcisrz n
izige 4 .. 2
242 cisz
1,0 2
22
2
k
kcisz
452,
42 ciscisz
iiz
21
212,
21
212 iiz 22,22
y
x
OR2
4cis
324
cis
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4 16 0ii x
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4 16 0ii x 4 16x 4 16 0x cis
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4 16 0ii x 4 16x 4 16 0x cis
22 0,1,2,34kx cis k
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4 16 0ii x 4 16x 4 16 0x cis
22 0,1,2,34kx cis k
32 0,2 ,2 ,22 2
x cis cis cis cis
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4 16 0ii x 4 16x 4 16 0x cis
22 0,1,2,34kx cis k
32 0,2 ,2 ,22 2
x cis cis cis cis
2,2 , 2, 2x i i
![Page 24: X2 t01 09 de moivres theorem](https://reader033.vdocument.in/reader033/viewer/2022060116/557e77cdd8b42a4d108b4b69/html5/thumbnails/24.jpg)
4 16 0ii x 4 16x 4 16 0x cis
22 0,1,2,34kx cis k
32 0,2 ,2 ,22 2
x cis cis cis cis
2,2 , 2, 2x i i
y
x
OR
![Page 25: X2 t01 09 de moivres theorem](https://reader033.vdocument.in/reader033/viewer/2022060116/557e77cdd8b42a4d108b4b69/html5/thumbnails/25.jpg)
4 16 0ii x 4 16x 4 16 0x cis
22 0,1,2,34kx cis k
32 0,2 ,2 ,22 2
x cis cis cis cis
2,2 , 2, 2x i i
y
x
OR
2 0cis
![Page 26: X2 t01 09 de moivres theorem](https://reader033.vdocument.in/reader033/viewer/2022060116/557e77cdd8b42a4d108b4b69/html5/thumbnails/26.jpg)
4 16 0ii x 4 16x 4 16 0x cis
22 0,1,2,34kx cis k
32 0,2 ,2 ,22 2
x cis cis cis cis
2,2 , 2, 2x i i
y
x
OR
2 0cis
22
cis
2cis
22
cis
![Page 27: X2 t01 09 de moivres theorem](https://reader033.vdocument.in/reader033/viewer/2022060116/557e77cdd8b42a4d108b4b69/html5/thumbnails/27.jpg)
4 16 0ii x 4 16x 4 16 0x cis
22 0,1,2,34kx cis k
32 0,2 ,2 ,22 2
x cis cis cis cis
2,2 , 2, 2x i i
y
x
OR
2 0cis
22
cis
2cis
22
cis
Cambridge: Exercise 7A; 1, 2, 3 abef, 5, 6, 7,
9 to 14, 16 to 18
Patel: Exercise 4E; 1 to 4 ac