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(cos( )θ + j sin( ))θ n = cos( θ +n ) j sin( θn ) (6.27)

The above results suggest that the polar form of a complex number may be
written as a function of an exponential function because of the additivity of
the arguments upon multiplication. We revisit this issue later.

In-Class Exercises
z z 1
Pb. 6.11 Show that 1 = [cos(θ 1 − θ 2 ) + j sin(θ 1 − θ 2 . )]
z z
2 2
Pb. 6.12 Explain, using the above results, why multiplication of any com-
plex number by j is equivalent to a rotation of the point representing this
number in the complex plane by π/2.
Pb. 6.13 By what angle must we rotate the point P(3, 4) to transform it to the
point P′(4, 3)?

Pb. 6.14 The points z = 1 + 2j and z = 2 + j are adjacent vertices of a regular
1
2
hexagon. Find the vertex z that is also a vertex of the same hexagon and that
3
is adjacent to z (z ≠ z ).
2
1
3
Pb. 6.15 Show that the points A, B, C representing the complex numbers z ,
A
z , z in the complex plane lie on the same straight line if and only if:
B
C
z − z
A c is real.
z − z
B c
Pb. 6.16 Determine the coordinates of the P′ point obtained from the point
P(2, 4) through a reflection around the line y = x + 2.
2
Pb. 6.17 Consider two points A and B representing, in the complex plane,
the complex numbers z and 1/.z 1 Let P be any point on the circle of radius
1
1 and centered at the origin (the unit circle). Show that the ratio of the length
of the line segments PA and PB is the same, regardless of the position of point
P on the unit circle.
Pb. 6.18 Find the polar form of each of the following quantities:

(1+ j ) 15 2 3 99
)( +
, ( 1−+ jj 2 ), (1+ +j j + j )
(1− j ) 9

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