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 10 below.] When any of these functions is defined, however, Problem 37, Exercises for Chapter 2 Sections 3, 4, and 5 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 8 and 18 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.