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