22portal.unimap.edu.my/portal/page/portal30/Lecture Notes...Tutorial1Scheme/EQT101/20172018/ pg. 8...

Tutorial1Scheme/EQT101/20172018/ pg. 1

TUTORIAL 1: ANSWER SCHEME

Question 1

a)

2 2

1

2 3

2 3 2 3

2 3 2 3 13

3arg tan 56.31

2

z i

z i i

z i

z

b)

2

4

4 4

4 4 4

arg 90 / 270

z i

z i i

z i

z

c)

2

16

16 16

16 16 16

arg 0 / 360

z

z

z

z


d)

2 2

1

3

3 3

3 3 1 10

1arg tan 18.43 / 341.57

3

z i

z i i

z i

z

e)

2 2

1

1 5

1 5 1 5

1 5 1 5 26

1arg tan 101.31 / 258.69

5

z i

z i i

z i

z

f)

22

1

1 3

1 3 1 3

1 3 1 3 2

3arg tan 120

1

z i

z i i

z i

z


Question 2

a) 2 3 2 1 3p q i i i

b) 4 2 5 4 7z w i i i

c) 2 3 2 2 3 4 2 16 4p z i i i


d) 3 3 2 3 5 12 2q iw i i i i

Question 3

a) 2 5 3 15 6zw i i i

b) 2 5 3 5 2

3 3 3 3

z i ii

w i i

c) 3 3 2 5 15 6

2 5 2 5 29 292 5

w i i ii

i iz i

d)

3 3 2 5

2 5 2 5 2 5

15 6 15 6

29 29 29

w i i i

z i i i

ii

e) 4 2 5 4 2 5 3 5 2

3 3 3 3 3

z i i ii

w i i i

f) 5 5 3 2 5 5 31

2 5 2 5 29 29

w i ii

z i i

g)

22 2 5 3

2 53 3

58 58

3 3

z i iz i

w i i

i

Question 4

a)

5

1 2

5 3

3 3

3 1

2 2

pi i

i

i i

i

b)

2

3 2

4

3 2 15 8

15 8 15 8

29 54

289 289

iq

i

i i

i i

i


c)

7

3

4

2 3

2 3

1 2 3

2 3 2 3

2 3

13 13

iw

i i

i

i

i

i i

i

d) 4

5

1

5

1 1

5

1

5

is

i

i

i

e)

6 3

5 2 1

6 5 2 3 1

5 2 5 2 1 1

30 121 2

29 29

59 46

29 29

iz

i i

i i i

i i i i

i i

i

Question 5

a) 2 3iz z i

2 3

2 3

2 2 3

Re : Im :

2 2 3

6 3,

5 5

6 3

5 5

i a bi a bi i

i a bi a b i

b ai a b i

z z

b a a b

a b

z i

b) 5 42

z iz

5 42

15 4 4

2 2

Re : Im :

15 4 4

2 2

10 1,

9 9

10 1

9 9

a bi ia bi

a ba bi i

z z

a ba b

a b

z i


c) 2 2 5z i iz

2 2 5

3 2 5 5

Re : Im :

3 2 5 5

1 5,

14 14

1 5

14 14

a bi a bi i a bi

a bi b ai

z z

a b b a

a b

z i

d) 3

2z i

i z

32

3 2 2 1

Re : Im :

2 3 2 2

10,

3

1

3

a bi i

i a bi

a b i a b i

z z

a a b b

a b

z i

Question 6

a) 2z i

1

0.1476

5

1arg tan

2

26.57

0.1476

: 5 cos 26.57 sin 26.57

: 5 i

z

z

PF z i

EF z e

b) 3 2z i

1

0.8128 1.1872

13

2arg tan

3

146.31 / 213.69

0.8128 /1.1872

: 13 cos146.31 sin146.31

13 cos 213.69 sin 213.69

: 13 / 13i i

z

z

PF z i

z i

EF z e z e

c) 4 7z i

1

0.3348 1.6652

65

7arg tan

4

60.26 / 299.74

0.3348 /1.6652

: 65 cos60.26 sin 60.26

65 cos 299.74 sin 299.74

: 65 / 65i i

z

z

PF z i

z i

EF z e z e

d) 3z i

2

3

arg 90

2

: 3 cos90 sin 90

: 3i

z

z

PF z i

EF z e


Question 7

a)

6 cos30 sin 30

3 cos90 sin 90

iz

i

6 cos 30 90 sin 30 90

3

6 cos 120 sin 120

3

1 1

6 2

iz

i

i

b) 2 2 cos45 sin 45 cos180 sin180z i i

2 2 cos 45 180 sin 45 180

2 2 cos 135 sin 135

1 12 2

2 2

2 2

z i

i

i

i

c) 2

2

5

i

i

ez

e

2

2

2

5

2

5

i

i

z e

e

Transform the equation into the polar

form:

2

cos90 sin 905

2

5

z i

i

Write in the standard form:

2

5z i

d) 4 2

5 3.7i i

z e e

4 2

5 3

22

15

7

7

i

i

z e

e

Transform the equation into polar

form:

7 cos 264 sin 264

0.7315 6.9617

z i

i

Write in the standard form:

0.7315 6.9617z i


Question 8

a) 1

3 34 , ?z i z

11

33

4 , 17, arg 14.04

3, 0,1,2

14.04 360 14.04 36017 cos sin

3 3k

w i w w

n k

k kz i

0

11

330

1

3

1

3

0 :

14.04 14.0417 cos sin

3 3

17 cos 4.68 sin 4.68

17 0.9967 0.0816

1.5982 0.1308

1.6035z

k

z i

i

i

i

r

1

11

331

1

3

1

3

1:

14.04 360 14.04 36017 cos sin

3 3

17 cos 115.32 sin 115.32

17 0.4277 0.9039

0.6858 1.4495

1.6035z

k

z i

i

i

i

r

1

11

332

1

3

1

3

2 :

14.04 720 14.04 72017 cos sin

3 3

17 cos 235.32 sin 235.32

17 0.569 0.8224

0.9124 1.3187

1.6035z

k

z i

i

i

i

r


b) 1

2 22 3 , ?z i z

11

22

2 3 , 13, arg 123.69

2, 0,1

123.69 360 123.69 36013 cos sin

3 3k

w i w w

n k

k kz i

0

11

220

1

2

1

2

0 :

123.69 123.6913 cos sin

2 2

13 cos 61.85 sin 61.85

13 0.4719 0.8817

0.896 1.6741

1.5334z

k

z i

i

i

i

r

1

11

221

1

2

1

2

1:

123.69 360 123.69 36013 cos sin

2 2

13 cos 241.845 sin 241.845

13 0.4719 0.8817

0.896 1.6741

1.5334z

k

z i

i

i

i

r

c) 1

4 45 0, ?z i z

1 1

4 4

5 , 5, arg 90

4, 0,1,2,3

90 360 90 3605 cos sin

4 4k

w i w w

n k

k kz i


0

1 1

4 40

1

4

1

4

0 :

90 905 cos sin

4 4

5 cos 22.5 sin 22.5

5 0.9239 0.3827

1.3815 0.5722

1.4953z

k

z i

i

i

i

r

1

1 1

4 41

1

4

1

4

1:

90 360 90 3605 cos sin

4 4

5 cos 67.5 sin 67.5

5 0.3827 0.9239

0.5722 1.3815

1.4953z

k

z i

i

i

i

r

2

1 1

4 42

1

4

1

4

2 :

90 720 90 7205 cos sin

4 4

5 cos 157.5 sin 157.5

5 0.9239 0.3827

1.3815 0.5722

1.4953z

k

z i

i

i

i

r

3

1 1

4 43

1

4

1

4

3:

90 1080 90 10805 cos sin

4 4

5 cos 247.5 sin 247.5

5 0.3827 0.9239

0.5722 1.3815

1.4953z

k

z i

i

i

i

r


d) 1

3 32 0, ?z z

1 13 3

2, 2, arg 0

3, 0,1,2

0 360 0 3602 cos sin

3 3k

w w w

n k

k kz i

0

1 13 30

1

3

1

3

0 :

0 02 cos sin

3 3

2 cos 0 sin 0

2 1

1.2599

1.2599z

k

z i

i

r

1

1 13 3

1

1

3

1

3

1:

0 360 0 3602 cos sin

3 3

2 cos 120 sin 120

1 32

2 2

0.63 1.0911

1.2599z

k

z i

i

i

i

r

2

1 13 32

1

3

1

3

2 :

0 720 0 7202 cos sin

3 3

2 cos 240 sin 240

1 32

2 2

0.63 1.0911

1.2599z

k

z i

i

i

i

r


Question 9

Multiple angles n

Theorem 6:

12cos

12 sin

n

n

n

n

z nz

z i nz

a) 3cos

From theorem 6, let 1:n

1

2cos 1zz

Equation 1 to the power of 3:

3

312cos 2z

z

From (2):

RHS: 3 32cos 8cos

LHS:

3 2 3

3 3 2 3 3

1 2 3

3

3

3

3

1 1 1 1

1 13 3

1 13

2cos3 3 2cos

2cos3 6cos

z z C z C z Cz z z z

z zz z

z zz z

Equate LHS and RHS 3

3

8cos 2cos3 6cos

1 3cos cos3 cos

4 4

b) 4cos


1

2cos 3zz


4

412cos 4z

z


From (4):

RHS: 4 42cos 16 cos

LHS:

4 2 3 4

4 4 3 4 3 4 4

1 2 3 4

4 2

2 4

24

4 2

1 1 1 1 1

1 14 6 4

1 14 6

2cos 4 4 2cos 2 6

2cos 4 8cos 2 6

z z C z C z C z Cz z z z z

z zz z

z zz z


4

16cos 2cos 4 8cos 2 6

1 1 3cos cos 4 cos 2

8 2 8

c) 5cos


1

2cos 5zz


5

512cos 6z

z

From (6):

RHS: 5 52cos 32 cos

LHS:

5 2 3 4 5

5 5 4 5 3 5 2 5 5

1 2 3 4 5

4 3

3 5

35

5 3

1 1 1 1 1 1

1 1 15 10 10 5

1 1 15 10

2cos5 5 2cos3 10 2cos

2cos5 10cos 2 20cos

z z C z C z C z C z Cz z z z z z

z z zz z z

z z zz z z


5

32cos 2cos5 10cos3 20cos

1 5 5cos cos5 cos3 cos

16 16 8


d) 3sin


1

2 sin 7z iz


3

312 sin 8z i

z

From (8):

RHS: 3 32 sin 8 sini i

LHS:

3 2 3

3 3 2 3 3

1 2 3

3

3

3

3

1 1 1 1

1 13 3

1 13

2 sin 3 3 2 sin

2 sin 3 6 sin

z z C z C z Cz z z z

z zz z

z zz z

i i

i i


3

8 sin 2 sin 3 6 sin

1 3sin sin 3 sin

4 4

i i i

e) 4sin


1

2 sin 9z iz


4

412 sin 10z i

z

From (10):

RHS: 4 42 sin 16sini


LHS:

4 2 3 4

4 4 3 4 3 4 4

1 2 3 4

4 2

2 4

24

4 2

1 1 1 1 1

1 14 6 4

1 14 6

2cos 4 4 2cos 2 6

2cos 4 8cos 2 6

z z C z C z C z Cz z z z z

z zz z

z zz z


4

16sin 2cos 4 8cos 2 6

1 1 3sin cos 4 cos 2

8 2 8

f) 5sin


1

2 sin 11z iz


5

512 sin 12z i

z

From (12):

RHS: 5 52 sin 32 sini i

LHS:

5 2 3 4 5

5 5 4 5 3 5 2 5 5

1 2 3 4 5

5 3

3 5

35

5 3

1 1 1 1 1 1

1 1 15 10 10 5

1 1 15 10

2 sin 5 5 2 sin 3 10 2 sin

2 sin 5 10 sin 2 2

z z C z C z C z C z Cz z z z z z

z z zz z z

z z zz z z

i i i

i i

0 sini


5

32 sin 2 sin 5 10 sin 3 20 sin

1 5 5sin sin 5 sin 3 sin

16 16 8

i i i i


Question 10

De Moivre’s Theorem (DMT)

cos sin cos sinn

i n i n

a) From the DMT, when n =3:

3

cos sin cos3 sin 3 13i i

LHS, let:

cos

sin

a

b i

3 3

3 3 2 3 2 3 3

1 2 3

3 2 2 3

3 2 2 3

3 2 2 3

cos sin

3 3

cos 3 cos sin 3cos sin sin

cos 3cos sin 3cos sin sin

i a b

a C a b C ab C b

a a b ab b

i i

i

Equate LHS and RHS

3 2 2 3

3 2

2 3

cos3 sin 3 cos 3cos sin 3cos sin sin

Re : cos3 cos 3cos sin

Im : sin 3 3cos sin sin

i i

z

z

b) From the DMT, when n =4:

4

cos sin cos 4 sin 4 14i i

LHS, let:

cos

sin

a

b i

4 4

4 4 3 4 2 2 4 3 4 4

1 2 3 4

4 3 2 2 3 4

4 3 2 2 3 4

4 2 2 4 3 3

cos sin

4 6 4

cos 4 cos sin 6 cos sin 4 cos sin sin

cos 6 cos sin sin 4 cos sin 4 cos sin

i a b

a C a b C a b C ab C b

a a b a b ab b

i i

i


Equate LHS and RHS

4 2 2 4 3 3

4 2 2 4

3 3

cos 4 sin 4 cos 6 cos sin sin 4 cos sin 4 cos sin

Re : cos 4 cos 6 cos sin sin

Im : sin 4 4 cos sin 4 cos sin

i i

z

z

c) From the DMT, when n =5:

5


LHS, let:

cos

sin

a

b i

5 5

5 5 4 5 3 2 5 2 3 5 4 5 5

1 2 3 4 5

5 4 3 2 2 3 4 5

5 4 3 2 2 3 4 5

5 3 2 4 4 2

cos sin

5 10 10 5

cos 5 cos sin 10 cos sin 10 cos sin 5cos sin sin

cos 10 cos sin 5cos sin 5cos sin 10 cos si

i a b

a C a b C a b C a b C ab C b

a a b a b a b ab b

i i i

i

3 5n sin

Equate LHS and RHS

5 3 2 4 4 2 3 5

5 3 2 4

4 2 3 5

cos5 sin 5 cos 10 cos sin 5cos sin 5cos sin 10 cos sin sin

Re : cos5 cos 10 cos sin 5cos sin

Im : sin 5 5cos sin 10 cos sin sin

i i

z

z

d) From the DMT, when n =6:

6


LHS, let:

cos

sin

a

b i

6 6

6 6 5 6 4 2 6 3 3 6 2 4 6 5 6 6

1 2 3 4 5 6

6 5 4 2 3 3 2 4 5 6

6 5 4 2 3 3

2 4 5 6

6 4 2

cos sin

6 15 20 15 6

cos 6 cos sin 15cos sin 20 cos sin

15cos sin 6 cos sin sin

cos 15cos sin 15co

i a b

a C a b C a b C a b C a b C ab C b

a a b a b a b a b ab b

i i

i

2 4

5 3 3 5

s sin

6 cos sin 20 cos sin 6 cos sini


Equate LHS and RHS

6 4 2 2 4 6

5 3 3 5

6 4 2 2 4 6

5 3 3 5

cos6 sin 6 cos 15cos sin 15cos sin sin

6 cos sin 20 cos sin 6 cos sin

Re : cos6 cos 15cos sin 15cos sin sin

Im : sin 6 6 cos sin 20 cos sin 6 cos sin

i

i

z

z

Question 11

a) 4 2 3z i

2 2

2 2

4 2 3

4 2 3

4 2 3

4 2 9

x iy i

x i y

x y

x y

A circle with radius, r =3 and centre, C(4,-2).

b) 3 5z z

22 2 2

22 2 2

2 2 2 2

2 2

2 2

2

2

3 5

3 5

3 5

9 5

9 9 10 25

8 10 25 8 0

5 250

4 4

5 225

8 64

x iy x iy

x iy x iy

x y x y

x y x y

x y x x y

x x y

x x y

x y

A circle with radius, r = 15

8and centre,

5,0

8C

.


c) 3 2z i z

2 22 2

2 22 2

2 2 2 2

3 2

3 2

3 2

3 2

6 9 4 4

6 4 5

2 5

3 6

x iy i x iy

x y i x iy

x y x y

x y x y

x y y x x y

y x

y x

A

straight line with slope, 2

3m .

d) arg4

z i

1

arg4

arg 14

1tan

4

1tan

4

11

1

x iy i

x y i

y

x

y

x

y

x

y x

A half line from (0,1) with gradient, 1m

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