Maths Quest Maths C Year 12 for Queensland 2e 1

WorkSHEET 8.1 Advanced exponential functions

Name: ___________________________ 1 Using de Moivre’s theorem and the binomial

expansion, prove that ( ) 1cos2sincos2cos 222 -=-= qqqq

( )22 sincos

sincos

qq

qq

iziz+=

+=

Using de Moivre’s theorem, qq 2sin2cos2 iz +=

Writing the binomial expansion of ,2z we have

qqqqqqq

22

222

sincos2cossincos2sincos

-=\

+-= iz

Applying the Pythagorean Identity,

( )

1cos2sincos2cos1cos2cos1cos2cos

cos1sin

2

22

2

22

22

-=

-=\

-=

--=\

-=

qqqq

qqqq

qq

2 Using the multiple angle formulas, prove that ( ) ( ) ( ) ( )2sin cos = sin sinx x x x4 2 6 + 2

( ) ( )( )( )( )( )

xxizz

izz

zzzzi

zzzzi

zzzzi

zzzzi

xx

2sin6sin22

212121

21

212

2cos4sin2

2266

2266

6226

2244

2244

+=

-+

-=

-+-=

--+=

--=

-´-´=

--

--

--

--

--

Maths Quest Maths C Year 12 for Queensland 2e 2

3 Prove that ( ) ( ) ( )[ ]32cos44cos

81sin 4 +-= xxx

( )

( )( ) ( )

( )

( )

( )32cos44cos81

32cos44cos162

62cos84cos2161sin

62cos84cos262cos244cos264

.4.6.4

16sin

2sin

4

2244

432213441

414

1

+-=

+-=

+-=

+-=+´-=++-+=

+-+-=-

-=

-=

--

-----

-

-

xx

xx

xxx

xxxxzzzz

zzzzzzzzzz

zzx

izzx

4 Express 3

2

2i

ep

in standard form.

31

232

212

32sin2

32cos2

2 32

i

i

i

ei

+-=

´+÷øö

çèæ-´=

+=pp

p

Maths Quest Maths C Year 12 for Queensland 2e 3

5 If iu -= 3 and ,1 iw += (a) express both u and w in Euler’s form.

(b) express 3

5

uw

in standard form.

(c) find values for m and n such that

8m nu w i= .

iwiu +=-= 1,3 (a) 2=u

u is a complex number in the 4th quadrant of the complex plane

22

6arg

6

=

=\

-=\

-

weu

u

ip

p

w is in the first quadrant of the complex plane

42

4arg

i

ew

w

p

p

-=\

-=

(b)

i

i

e

e

e

e

ewu

i

i

i

i

i

+=

÷ø

öçè

æ+=

=

=

=

-

-

-

122

12222

24

8

2

2

4

43

2

45

25

23

5

3

p

p

p

p

p

(cont.)

Maths Quest Maths C Year 12 for Queensland 2e 4

5 (cont.)

(c)

23

23

1223

2

642

426

22

2 RHS

.2

.2

2.2

LHS8

nm

i

mninm

mninm

ninmim

nm

nm

e

e

e

ee

wuiwu

+

÷øö

çèæ -

+

÷øö

çèæ -+

-

=\

=

=

=

=

=

=

p

p

p

pp

.integer an for 221

1223 and

622

3 i.e.

kkmnnm

nm

+=-

=+

+=

kmkmk

nmknkn

mnkmn

35.1i.e.632

63662

6324124

62 and24623 i.e.

-=-=

--=-=\+=+=\

=++=-

There is an infinite solution set given by

.integer for 63and 35.1

kknkm

+=-=

Maths Quest Maths C Year 12 for Queensland 2e 5

6 Apply Euler’s formula to evaluate

( )sin dxe x x2ò

( )

( )( )

( )( )( )

( )

( )

( ) ( )

( )

( )( )

2

2

2

2

1 2

1 2

1 2

1 2

2

sin 2 d

cos 2 sin 2

Im sin 2

sin 2 d

Im d

Im . d

Im d

Im1 2

1 2Im1 2 1 2

1 2Im

5

Im . 1 25

Im cos 2 sin 2 1 25

x

ix

ix

x

x ix

x ix

i x

i x

i x

i x

xix

x

e x x

e x i x

e x

e x x

e e x

e e x

e x

ei

e ii i

e i

e e i

e x i x i

+

+

+

+

= +

=

\

=

=

=

é ù= ê ú+ë û

é ù-= ´ê ú+ -ë û

é ù-= ê ú

ê úë ûé ù

= -ê úë ûé

= + -

ò

òòòò

( )

( )

( )

( )

cos 2 2 cos 2Im

sin 2 2sin 25

cos 2 2sin 25Im

sin 2 2cos 25

sin 2 d

sin 2 2cos 25

x

x

x

x

x

x i xei x x

e x x

ie x x

e x x

e x x c

ùê úë ûé + ùæ ö

= ê úç ÷+ +è øë ûé ù

+ê úê ú=ê ú

+ -ê úë û

\ =

= - +

ò

Maths Quest Maths C Year 12 for Queensland 2e 6

7 (a) Sketch the function ( )= cosxy e x- over the domain xp p- £ £ .

(b) Determine ( )xe x

xcoslim -

¥®

(c) Evaluate ( )ò¥

-

0

cos dxxe x

(a) Here, xcos is squeezed between the envelopes .xe-± Graph xey -±= and then squeeze

xy cos= between the envelopes

(b) ( ) 0coslim =- xe x

xe- converges rapidly to zero while xcos oscillates between .1±

Hence ( )xe x cos- oscillates towards 0=y as x increases.

(c) ò¥

-

0

dcos xxe x

formula. parts Apply the

dcosConsider ò -= xxeI x

( )xxexe

xxex

xe

xxx

eI

xx

xx

x

dsinsin

dsinddsin

dsindd

ò

ò

ò

--

--

-

+=

-=

÷øö

çèæ=

Now consider ò - xxe x dsin

*** It is interesting that the book uses Integration by Parts here, instead of converting to Euler form … I would have thought using the new integration technique would be easier and quicker … ??? ***

(cont.)

Maths Quest Maths C Year 12 for Queensland 2e 7

7 (cont.)

( )

( )

( )xxeI

xxe

xexeI

IxexeI

Ixe

xxexe

xxx

exxe

x

x

xx

xx

x

xx

xx

cossin21

cossin

cossin2

cossin i.e.

cos

dcoscos

dcosdddsin

-=\

-=

-=\

--=

--=

--=

-=

-

-

--

--

-

--

--

ò

òò

.

Hence,

( )

( ) ( )( )

.21

1210

0cos0sin21

cossin21lim

d cos

dcoslimdcos

0

0

00

=

-´-=

þýü--

îíì -=

\

=

-

-

¥®

¥-

-

¥®

¥-

ò

òò

e

nne

xxe

xxexxe

nn

x

nx

nx

*** Not really sure that Limit theory is required here … we are just doing a definite integral … ??? … just doing a standard definite integral process here gets the correct answer … J

Maths Quest Maths C Year 12 for Queensland 2e 8

8 (a) Find the set of complex numbers where n = 1, 2 and 3 given that

(b) Graph in the complex plane joining

the points together to form a closed figure. What shape is this figure?

(a)

(b) Plot each point in the sequence in the

complex plane. Join them together. The figure formed is an equilateral triangle.

Maths Quest Maths C Year 12 for Queensland 2e 9

9 Show that

10 Given that represents the displacement of a particle at time t, (a) show that

(b) If show that

(c) By making appropriate use of graphics

calculator functions, find the first positive

value of a such that .

(a)

(b)

(c) Solve

(Hint: Use a ‘solver’ routine in the graphics calculated to show that )