30 Chapter^1
Since the trigonometric functions cos () and sin() can be defined analytically
by means of the power series expansions,
()2 ()4
cos()= 1-- + - - .. ·
2! 4!
()3 ()5
sin()=() - - + - - .. ·
3! 5!the formulas (1.7-5) can be used to introduce analytically the argument()=
arg z as the set of solutions of the system (1.7-5). In particular, () 0 = Arg z
is that solution of (1.7-5) which lies in the interval (-7r, 7r]; alternatively,
()1 = Argz is the solution of (1.7-5) which lies in the interval [0,27r). That
there is just one solution in each case is clear from the discussion of the
real functions sin() and cos () in trigonometry, or in introductory courses
on real variables.
1.8 Polar Form of the Complex Number
The modulus r and the argument () of a nonzero complex number may be
thought as the polar coordinates of its affix P. From (1. 7-5) we have
a= r cos() and b = r sin() (1.8-1)
Hence the complex number z = a + bi may be written in the form
z = r( cos () + i sin()) (1.8-2)
which is called the polar (or trigonometric) form of the complex number.
If eo denotes the principal argument, we have () = ()0 + 2k7r, and
sin() = sin( 80 + 2h) = sin () 0
cos () = cos( 80 + 2k7r) = cos () 0Thus it is immaterial what value of the argument is chosen in_ the
representation (1.8-2).
The polar form of the complex number is sometimes abbreviated by
writing z = r cis ().
Examples 1. If z = i, we have r = I and () 0 = 7r /2. Hence
. 7r.. 7r
z = cos - + z sm -
2 2