x2 t01 05 de moivres theorem
TRANSCRIPT
![Page 1: X2 T01 05 de moivres theorem](https://reader034.vdocument.in/reader034/viewer/2022052621/557e77e9d8b42a4d108b4b95/html5/thumbnails/1.jpg)
De Moivre’s Theorem
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De Moivre’s Theorem
nnini n
integers positive allfor sincossincos
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De Moivre’s Theorem
nnini n
integers positive allfor sincossincos
this extends to;
![Page 4: X2 T01 05 de moivres theorem](https://reader034.vdocument.in/reader034/viewer/2022052621/557e77e9d8b42a4d108b4b95/html5/thumbnails/4.jpg)
De Moivre’s Theorem
nnini n
integers positive allfor sincossincos
ninrir nn sincossincos
this extends to;
![Page 5: X2 T01 05 de moivres theorem](https://reader034.vdocument.in/reader034/viewer/2022052621/557e77e9d8b42a4d108b4b95/html5/thumbnails/5.jpg)
De Moivre’s Theorem
nnini n
integers positive allfor sincossincos
ninrir nn sincossincos
this extends to;
51 .. iige
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De Moivre’s Theorem
nnini n
integers positive allfor sincossincos
ninrir nn sincossincos
this extends to;
51 .. iige 2
11 22
z
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De Moivre’s Theorem
nnini n
integers positive allfor sincossincos
ninrir nn sincossincos
this extends to;
51 .. iige 2
11 22
z
4
11tanarg 1
z
![Page 8: X2 T01 05 de moivres theorem](https://reader034.vdocument.in/reader034/viewer/2022052621/557e77e9d8b42a4d108b4b95/html5/thumbnails/8.jpg)
De Moivre’s Theorem
nnini n
integers positive allfor sincossincos
ninrir nn sincossincos
5
42
cis
this extends to;
51 .. iige 2
11 22
z
4
11tanarg 1
z
![Page 9: X2 T01 05 de moivres theorem](https://reader034.vdocument.in/reader034/viewer/2022052621/557e77e9d8b42a4d108b4b95/html5/thumbnails/9.jpg)
De Moivre’s Theorem
nnini n
integers positive allfor sincossincos
ninrir nn sincossincos
5
42
cis
this extends to;
51 .. iige 2
11 22
z
452 5 cis
4
11tanarg 1
z
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De Moivre’s Theorem
nnini n
integers positive allfor sincossincos
ninrir nn sincossincos
5
42
cis
this extends to;
51 .. iige 2
11 22
z
452 5 cis
4
11tanarg 1
z
4324 cis
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43sin
43cos24 i
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43sin
43cos24 i
i
21
2124
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43sin
43cos24 i
i
21
2124
i44
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43sin
43cos24 i
i
21
2124
i44
sinandcos ofin terms2sinand2cos Express ii
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43sin
43cos24 i
i
21
2124
i44
sinandcos ofin terms2sinand2cos Express ii
2sincos2sin2cos ii
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43sin
43cos24 i
i
21
2124
i44
sinandcos ofin terms2sinand2cos Express ii
2sincos2sin2cos ii 22 sincossin2cos i
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43sin
43cos24 i
i
21
2124
i44
sinandcos ofin terms2sinand2cos Express ii
2sincos2sin2cos ii 22 sincossin2cos i
By equating real and imaginary parts
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43sin
43cos24 i
i
21
2124
i44
22 sincos2cos
sinandcos ofin terms2sinand2cos Express ii
2sincos2sin2cos ii 22 sincossin2cos i
By equating real and imaginary parts
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43sin
43cos24 i
i
21
2124
i44
22 sincos2cos
sinandcos ofin terms2sinand2cos Express ii
2sincos2sin2cos ii 22 sincossin2cos i
cossin22sin
By equating real and imaginary parts
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Finding Roots
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Finding Rootsrcisziyxz
n
n
If
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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
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Finding Rootsrcisziyxz
n
n
If
1,,1,0 2
nknkcisrz n
izige 4 .. 2
242 cisz
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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
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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
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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
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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
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08 4 xii
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08 4 xii84 x
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08 4 xii84 x
084 cisx
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08 4 xii84 x
084 cisx
3,2,1,0 4
284
kkcisx
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08 4 xii84 x
084 cisx
3,2,1,0 4
284
kkcisx
238,8,
28,08 4444 ciscisciscisx
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08 4 xii84 x
084 cisx
3,2,1,0 4
284
kkcisx
238,8,
28,08 4444 ciscisciscisx
iix 4444 8,8,8,8
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08 4 xii84 x
084 cisx
3,2,1,0 4
284
kkcisx
238,8,
28,08 4444 ciscisciscisx
iix 4444 8,8,8,8
nn
nn
zz
zziziii 1 and 1 find ,sincos If
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08 4 xii84 x
084 cisx
3,2,1,0 4
284
kkcisx
238,8,
28,08 4444 ciscisciscisx
iix 4444 8,8,8,8
nn
nn
zz
zziziii 1 and 1 find ,sincos If
ninzn sincos
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08 4 xii84 x
084 cisx
3,2,1,0 4
284
kkcisx
238,8,
28,08 4444 ciscisciscisx
iix 4444 8,8,8,8
nn
nn
zz
zziziii 1 and 1 find ,sincos If
ninzn sincos
ninzz
nn sincos1
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08 4 xii84 x
084 cisx
3,2,1,0 4
284
kkcisx
238,8,
28,08 4444 ciscisciscisx
iix 4444 8,8,8,8
nn
nn
zz
zziziii 1 and 1 find ,sincos If
ninzn sincos
ninzz
nn sincos1
nin sincos
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08 4 xii84 x
084 cisx
3,2,1,0 4
284
kkcisx
238,8,
28,08 4444 ciscisciscisx
iix 4444 8,8,8,8
nn
nn
zz
zziziii 1 and 1 find ,sincos If
ninzn sincos
ninzz
nn sincos1
nin sincos
xxxx
sinsinfunction odd is sincoscosfunctioneven is cos
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08 4 xii84 x
084 cisx
3,2,1,0 4
284
kkcisx
238,8,
28,08 4444 ciscisciscisx
iix 4444 8,8,8,8
nn
nn
zz
zziziii 1 and 1 find ,sincos If
ninzn sincos
ninzz
nn sincos1
nin sincos
xxxx
sinsinfunction odd is sincoscosfunctioneven is cos
nz
z nn cos21
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08 4 xii84 x
084 cisx
3,2,1,0 4
284
kkcisx
238,8,
28,08 4444 ciscisciscisx
iix 4444 8,8,8,8
nn
nn
zz
zziziii 1 and 1 find ,sincos If
ninzn sincos
ninzz
nn sincos1
nin sincos
xxxx
sinsinfunction odd is sincoscosfunctioneven is cos
nz
z nn cos21
niz
z nn sin21
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niv cos of in terms cos Express 3
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niv cos of in terms cos Express 3
3
3 1cos2
zz
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niv cos of in terms cos Express 3
3
3 1cos2
zz
333 133cos8
zzzz
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niv cos of in terms cos Express 3
3
3 1cos2
zz
333 133cos8
zzzz
zz
zz 131
33
![Page 46: X2 T01 05 de moivres theorem](https://reader034.vdocument.in/reader034/viewer/2022052621/557e77e9d8b42a4d108b4b95/html5/thumbnails/46.jpg)
niv cos of in terms cos Express 3
3
3 1cos2
zz
333 133cos8
zzzz
zz
zz 131
33
cos63cos2
![Page 47: X2 T01 05 de moivres theorem](https://reader034.vdocument.in/reader034/viewer/2022052621/557e77e9d8b42a4d108b4b95/html5/thumbnails/47.jpg)
niv cos of in terms cos Express 3
3
3 1cos2
zz
333 133cos8
zzzz
zz
zz 131
33
cos63cos2
cos433cos
41cos3
![Page 48: X2 T01 05 de moivres theorem](https://reader034.vdocument.in/reader034/viewer/2022052621/557e77e9d8b42a4d108b4b95/html5/thumbnails/48.jpg)
Alternative for finding iyx
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Alternative for finding iyx
ibaiyx If
![Page 50: X2 T01 05 de moivres theorem](https://reader034.vdocument.in/reader034/viewer/2022052621/557e77e9d8b42a4d108b4b95/html5/thumbnails/50.jpg)
Alternative for finding iyx
ibaiyx If
2then
iyxxa
![Page 51: X2 T01 05 de moivres theorem](https://reader034.vdocument.in/reader034/viewer/2022052621/557e77e9d8b42a4d108b4b95/html5/thumbnails/51.jpg)
Alternative for finding iyx
ibaiyx If
2then
iyxxa
ayb
2
![Page 52: X2 T01 05 de moivres theorem](https://reader034.vdocument.in/reader034/viewer/2022052621/557e77e9d8b42a4d108b4b95/html5/thumbnails/52.jpg)
Alternative for finding iyx
ibaiyx If
2then
iyxxa
ayb
2
ige 1612 Find ..
![Page 53: X2 T01 05 de moivres theorem](https://reader034.vdocument.in/reader034/viewer/2022052621/557e77e9d8b42a4d108b4b95/html5/thumbnails/53.jpg)
Alternative for finding iyx
ibaiyx If
2then
iyxxa
ayb
2
ige 1612 Find ..
2016121612 22
i
![Page 54: X2 T01 05 de moivres theorem](https://reader034.vdocument.in/reader034/viewer/2022052621/557e77e9d8b42a4d108b4b95/html5/thumbnails/54.jpg)
Alternative for finding iyx
ibaiyx If
2then
iyxxa
ayb
2
ige 1612 Find ..
2016121612 22
i
228
22012
a
![Page 55: X2 T01 05 de moivres theorem](https://reader034.vdocument.in/reader034/viewer/2022052621/557e77e9d8b42a4d108b4b95/html5/thumbnails/55.jpg)
Alternative for finding iyx
ibaiyx If
2then
iyxxa
ayb
2
ige 1612 Find ..
2016121612 22
i
228
22012
a
44
16
b
![Page 56: X2 T01 05 de moivres theorem](https://reader034.vdocument.in/reader034/viewer/2022052621/557e77e9d8b42a4d108b4b95/html5/thumbnails/56.jpg)
Alternative for finding iyx
ibaiyx If
2then
iyxxa
ayb
2
ige 1612 Find ..
2016121612 22
i
228
22012
a
44
16
b
ii 421612
![Page 57: X2 T01 05 de moivres theorem](https://reader034.vdocument.in/reader034/viewer/2022052621/557e77e9d8b42a4d108b4b95/html5/thumbnails/57.jpg)
Exercise 4D; evens
Exercise 4E; 1 to 4 ac
Exercise 4F; 1 to 4, 5ab, 10