be
any real number. To obtain the trigonometric functions of
one finds r > 0, a, and b such that 
and then uses the definitions:
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Chapter III
Trigonometry Using Complex Numbers
1. The Trigonometric Functions
Let be
any real number. To obtain the trigonometric functions of
one finds r > 0, a, and b such that
and then uses the definitions:
The above three letter functions are abbreviations for cosine, sine,
tangent, secant, cosecant, and cotangent, respectively. We note that the
requirement r > 0 insures that cosine and sine are defined for all
real numbers 
but the other four functions will be undefined for values of
which cause a zero in the denominator. [See Problem
4 below.] When any of these functions is defined, however, Problem
37, Exercises for Chapter 2 Sections 4, 5, and 6 shows that the definition
is not ambiguous; that is, the functions are well defined.
For example, since
we have
and the other three functions are the reciprocals of these. The same
results would be obtained from 
or any other nonzero complex number with 
as argument.
The definitions show that
The definitions also imply that 
Then
For complex numbers of absolute value 1, i.e., for r = 1, this becomes the Euler Formula
Replacing
with 
we get 
Taking
conjugates of each side of the Euler Formula, we get 
These equations show that 
and 
Example 1. Double angle formulas for cosine and sine.
We use the Euler Formula to obtain 
and 
in terms
of 
and 
as
follows:
Equating the real and imaginary parts on each side of the equation, we have
and
These are the double angle formulas for cosine and sine respectively.
Similarly one can derive half angle formulas for 
and 
in terms
of 
using
the fact that 
is one of the two square roots of 
[See Problems 6 and 14
below.]
Example 2. Addition Formulas for cosine and sine.
We use the Euler Formula to express 
and 
in terms
of 
and 
as follows:
If we now equate the real and imaginary parts on each side of the equation, we get
and
These are the addition formulas for the cosine and the sine, respectively. A symbolic aid for remembering these formulas is
(C + iS)(c + is) = (Cc - Ss) + i(Sc + Cs).
Example 3. Subtraction formula for tangent.
We seek 
in terms of 
and 
First
we note that
What we need here is only that there is a real number k such
that 
Taking
conjugates, we have 
Thus
It now follows from the definition of the tangent that
This is the subtraction formula for the tangent.
Do NOT use your calculator for the exercises in this section or in Section
2.
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