ONLINE: MATHEMATICS EXTENSION 2
Topic 2
COMPLEX NUMBERS
EXERCISE p2301
p001
Prove the following relationships
1
cos4      cos  4   4 cos  2   3
8
1
sin 4      cos  4   4 cos  2   3
8
physics.usyd.edu.au/teach_res/hsp/math/math.htm
p2301
1
p002
Show that


 1  i 3 
3  i  1  i 3 
3i
6
4
4
3
8
p003
Find the cubic roots of 2. Plot the roots on an Argand diagram.
p004
Find the fourth roots of 3  2 i . Plot the roots on an Argand diagram.
physics.usyd.edu.au/teach_res/hsp/math/math.htm
p2301
2
p005
Prove the following
cos  
1 i  i 
e  e 
2
sin  
1
 e i   ei  
2i
 e i   ei  
tan   i  i  i  
e e 
p006
Using the results of exercise (5) show
1  cos 
sin 

sin 
1  cos 
1  cos 
sin  / 2   
2
tan  / 2  
cos  / 2   
1  cos 
2
p007
Solve the equation z 5  1  0
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p2301
3
ANSWERS
a001
(a)
Prove the following relationships
1
cos4      cos  4   4 cos  2   3
8
1
sin 4      cos  4   4 cos  2   3
8
i 2
cos    i sin     e    cos  2   i sin  2 
2
cos    i sin     cos2    sin 2     i 2 cos   sin   
Equating real and imaginary parts
2
cos  2   cos2    sin 2  
sin  2   2 cos   sin  
Binomial Theorem: a very useful formula to remember for the expansion a function of
the form (a + b)n where n = 1, 2, 3, …
 a  b
n
 an 
n  n 1 1 n(n  1)  n 2  2 n(n  1)( n  2)  n 3 3
a
b 
a b 
a b 
1!
2!
3!
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p2301
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(4)(3) 2 2 (4)(3)(2) 3 (4)(3)(2)(1) 4
a b 
ab 
b
(2)(1)
(3)(2)(1)
(4)(3)(2)(1)
 a  b
4
 a 4  4 a 3 b1 
 a  b
4
 a 4  4 a 3 b1  6 a 2 b2  4 a b3  b4
cos    i sin     e
4
i  4 
 cos  4   i sin  4 
cos    i sin   
 cos4    (i )(4) cos3   sin    (i ) 2 (6) cos 2   sin 2    (i) 3 (4) cos   sin 3    ( i) 4 sin 4  
4
 cos4    sin 4    (6) cos2   sin 2     i (4) cos3   sin    (4) cos   sin 3   
Equating real and imaginary parts
cos  4   cos4    sin 4    (6) cos 2   sin 2  
sin  4   (4) cos3   sin    (4) cos   sin 3  
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cos  4   4 cos  2   3
 cos4    sin 4    (6) cos 2   sin 2  
 4 cos2    sin 2     3
 cos4    1  cos     6cos 2   1  cos 2   
2
 8cos2    1
 8cos4  
1
cos4      cos  4   4 cos  2   3
8
cos4    1  sin 2     1  2sin 2    sin 4  
2
 cos2    sin 2    2sin 2    sin 4  
 cos2    sin 2    sin 4  
 cos  2   sin 4  
1
sin 4      cos  4   4 cos  2   3
8
physics.usyd.edu.au/teach_res/hsp/math/math.htm
p2301
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a002
Show that




 1  i 3 
3  i  1  i 3 
3i
 1  i 3 
3  i  1  i 3 
3i
6
4
4
3
6
4
4
3
8
8
z1  3  i  2 e i  / 6
z16  26 e i 
z2  1  i 3  2 e i  / 3
z 2 4  2 4 e i 4 / 3
z3  3  i  2 e  i  / 6
z34  2 4 e i 2 / 3
z4  1  3i  2 e  i  / 3
z33  2 3 e i 
z1 z 2 z3 z4  26 443 e i   4 / 3 2 / 3   23 e i   4 / 32 / 3   8
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a003
Find the cubic root of 2. Plot the roots on an Argand diagram.
3
2R  ?
z  2 cos  2   i sin  2  
 2 cos  2 k   i sin  2 k   k  0, 1, 2
wk  z1/ 3  21/ 3 cos  2 k / 3  i sin  2 k / 3 
w0  21/ 3 cos  0   i sin  0    21/ 3

 1/ 2  i 

w1  21/ 3 cos  2 / 3  i sin  2 / 3   21/ 3  1/ 2  i

w2  21/ 3 cos  4 / 3  i sin  4 / 3   21/ 3
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p2301

3 / 2 

3/2 

8
a004
Find the fourth roots of 3  2 i . Plot the roots on an Argand diagram.
There are four roots.
4
3 2 i  ?
z  3 2 i
z  32  22  131/ 2
  arg  z   atan(2 / 3)  0.5880 rad
z  131/ 2 cos    i sin     131/ 2 e i   131/ 2 e i  2 k 
wk  z1/ 4  131/ 8 e
i  / 4  k / 2 
k  0, 1, 2, 3
 / 4  0.1651 rad
w0  131/ 8 cos  / 4   i sin  / 4    131/ 8 0.9892  i  0.1465 
w1  131/ 8 cos  / 4   / 2   i sin  / 4   / 2    131/ 8  0.1465  i  0.9892  
w2  131/ 8 cos  / 4     i sin  / 4      131/ 8  0.9892  i  0.1465 
w3  131/ 8 cos  / 4  3 / 2   i sin  / 4  3 / 2    131/ 8 0.1465  i  0.9892  
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p2301
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a005
Prove the following
cos  
1 i  i 
e  e 
2
sin  
e i   cos   i sin 
e i   e  i   2 cos 
e i   e  i   2i sin 
tan  
1
 e i   ei  
2i
 e i   ei  
tan   i  i  i  
e e 
e  i   cos   i sin 
1
cos    e i   e  i  
2
1
sin  
e i   e i  

2i
 e i   ei  
sin 
 i  i   i  
cos 
e e 
physics.usyd.edu.au/teach_res/hsp/math/math.htm
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a006
Using the results of exercise (5) show
1  cos 
sin 

sin 
1  cos 
1  cos 
sin  / 2   
2
tan  / 2  
cos  / 2   
tan  
1  cos 
2
 e i   ei  
sin 
 i  i  i  
cos 
e e 
Replace  by  /2
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 e i  / 2   e  i  / 2  
 e i  / 2   e  i  / 2    e i  / 2  
tan  / 2   i  i  / 2   i  / 2    i  i  / 2   i  / 2    i  / 2  
e
e
e

e
e

 e i1
z
 i  i     i 1
z2
 e 1
i
z1  e  1  (cos   1)  i sin 
z2  e i   1  (cos   1)  i sin 
z
z2  e i   1  (cos   1)  i sin 
z1 z1 z2

z2 z2 z2
z2 z2  (cos   1) 2  sin 2   cos 2   2 cos   sin 2   2 1  cos  
 (cos   1)  i sin    (cos   1)  i sin  
2 1  cos  
(cos   1) (cos   1)  sin 2   i  (cos   1) sin   (cos   1) sin  

2 1  cos  
cos2   1  sin 2   i   sin  cos   sin   sin  cos   sin  

2 1  cos  
z
i sin 
1  cos 
tan  / 2   i z

tan  / 2  
sin 
1  cos 
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sin 
 sin    1  cos  



1  cos   1  cos    1  cos  
sin  1  cos   sin  1  cos  


1  cos2 
sin 2 
1  cos  

sin 
tan  / 2  
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tan 2  / 2  
1  cos  
2
sin 2 
sin 2  / 2 
1  cos  

2
cos  / 2 
sin 2 
2
 1  cos   2 
 1  cos   2 
2
2
sin  / 2   
 cos  / 2   
 1  sin  / 2  
2
2
 sin  
 sin  
2
2
2

1  cos     1  cos   

sin  / 2  1 


sin 2    sin 2  

2
sin 2  / 2   sin 2   1  2 cos   cos 2    1  cos  
2
1  cos    1  cos    1  cos2  / 2
sin  / 2  
 
2 1  cos  
2
1  cos  
sin  / 2   
2
2
2
cos  / 2   
1  cos  
2
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a007
Solve the equation z 5  1  0
There are five roots for z.
z5  1  0
z 5  1  cos  2 k   i sin  2 k 
k  0,1, 2,3, 4
zk  cos  2 k / 5  i sin  2 k / 5
z0  1
z1  cos  2 / 5  i sin  2 / 5
z2  cos  4 / 5  i sin  4 / 5
z3  cos  6 / 5  i sin  6 / 5
z4  cos 8 / 5  i sin 8 / 5
When these five complex numbers are plotted on an Argand diagram, they will lie on
the circle x2 + y2 = 1 and be equally spaced with angular separation equal to 2/5 rad =
72o.
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