worksheet 8.1 advanced exponential functions...maths quest maths c year 12 for queensland 2e 1...

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 sincossincos

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

+=

-+

-=

-+-=

--+=

--=

-´-´=

--

--

--

--

--


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


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.)


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

+=-=


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


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.)


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


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.


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 )

worksheet 8.1 advanced exponential functions...maths quest maths c year 12 for queensland 2e 1...

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