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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
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
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 )