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De Moivre’s Theorem
De Moivre’s Theorem
nnini n
integers positive allfor sincossincos
De Moivre’s Theorem
nnini n
integers positive allfor sincossincos
this extends to;
De Moivre’s Theorem
nnini n
integers positive allfor sincossincos
ninrir nn sincossincos
this extends to;
De Moivre’s Theorem
nnini n
integers positive allfor sincossincos
ninrir nn sincossincos
this extends to;
51 .. iige
De Moivre’s Theorem
nnini n
integers positive allfor sincossincos
ninrir nn sincossincos
this extends to;
51 .. iige 2
11 22
z
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
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
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
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
43sin
43cos24 i
43sin
43cos24 i
i
21
2124
43sin
43cos24 i
i
21
2124
i44
43sin
43cos24 i
i
21
2124
i44
sinandcos ofin terms2sinand2cos Express ii
43sin
43cos24 i
i
21
2124
i44
sinandcos ofin terms2sinand2cos Express ii
2sincos2sin2cos ii
43sin
43cos24 i
i
21
2124
i44
sinandcos ofin terms2sinand2cos Express ii
2sincos2sin2cos ii 22 sincossin2cos i
43sin
43cos24 i
i
21
2124
i44
sinandcos ofin terms2sinand2cos Express ii
2sincos2sin2cos ii 22 sincossin2cos i
By equating real and imaginary parts
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
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
Finding Roots
Finding Rootsrcisziyxz
n
n
If
Finding Rootsrcisziyxz
n
n
If
1,,1,0 2
nknkcisrz n
Finding Rootsrcisziyxz
n
n
If
1,,1,0 2
nknkcisrz n
izige 4 .. 2
Finding Rootsrcisziyxz
n
n
If
1,,1,0 2
nknkcisrz n
izige 4 .. 2
242 cisz
Finding Rootsrcisziyxz
n
n
If
1,,1,0 2
nknkcisrz n
izige 4 .. 2
242 cisz
1,0 2
22
2
k
kcisz
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
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
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
08 4 xii
08 4 xii84 x
08 4 xii84 x
084 cisx
08 4 xii84 x
084 cisx
3,2,1,0 4
284
kkcisx
08 4 xii84 x
084 cisx
3,2,1,0 4
284
kkcisx
238,8,
28,08 4444 ciscisciscisx
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
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
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
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
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
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
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
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
niv cos of in terms cos Express 3
niv cos of in terms cos Express 3
3
3 1cos2
zz
niv cos of in terms cos Express 3
3
3 1cos2
zz
333 133cos8
zzzz
niv cos of in terms cos Express 3
3
3 1cos2
zz
333 133cos8
zzzz
zz
zz 131
33
niv cos of in terms cos Express 3
3
3 1cos2
zz
333 133cos8
zzzz
zz
zz 131
33
cos63cos2
niv cos of in terms cos Express 3
3
3 1cos2
zz
333 133cos8
zzzz
zz
zz 131
33
cos63cos2
cos433cos
41cos3
Alternative for finding iyx
Alternative for finding iyx
ibaiyx If
Alternative for finding iyx
ibaiyx If
2then
iyxxa
Alternative for finding iyx
ibaiyx If
2then
iyxxa
ayb
2
Alternative for finding iyx
ibaiyx If
2then
iyxxa
ayb
2
ige 1612 Find ..
Alternative for finding iyx
ibaiyx If
2then
iyxxa
ayb
2
ige 1612 Find ..
2016121612 22
i
Alternative for finding iyx
ibaiyx If
2then
iyxxa
ayb
2
ige 1612 Find ..
2016121612 22
i
228
22012
a
Alternative for finding iyx
ibaiyx If
2then
iyxxa
ayb
2
ige 1612 Find ..
2016121612 22
i
228
22012
a
44
16
b
Alternative for finding iyx
ibaiyx If
2then
iyxxa
ayb
2
ige 1612 Find ..
2016121612 22
i
228
22012
a
44
16
b
ii 421612
Exercise 4D; evens
Exercise 4E; 1 to 4 ac
Exercise 4F; 1 to 4, 5ab, 10