Principles of Mathematics in Operations Research

13.5 Trigonometric Functions 183

We assert that there exists positive numbers x such that C(x) = 0. Let XQ be

the smallest among them. We define number 7r by

7T = 2XQ.

Then, C(§) = 0, and S(§) = ±1. Since C(x) > 0 in (0, §), S is increasing in

(0, §); hence 5(f) = 1. Therefore,

and the addition formula gives

E(iri) = -1, E(2m) = 1;

hence

£(z + 27™) = £:(z),V2eC.
Theorem 13.5.1 The following are true:
(a) The function E is periodic, with period 2-Ki.
(b) The functions C and S are periodic, with period 2n.
(c) If0<t< 2ir, then E{it) ^ 1.
(d) IfzeCB \z\ = 1, 3 unique t e [0, 2TT) 3 E{it) = z.
Remark 13.5.2 The curve 7 defined by 7(f) = E(it), 0 < t < 2w is a
simple closed curve whose range is the unit circle in the plane. Since -f'(t) —
iE(it), the length of 7 is JQ \j'(t)\ dt = 2w. This is the expected result for
the circumference of a circle with radius 1.
The point ^(t) describes a circular arc of length to as t increases from 0
to to. Consideration of the triangle whose vertices are z\ — 0, Z2 = j(to)>
and Z3 = C(to) shows that C(t) and S(t) are indeed identical with cos(t) and
sin(t) respectively, the latter are defined as ratios of sides of a right triangle.

The saying the complex field is algebraically complete means that every
nonconstant polynomial with complex coefficients has a complex root.

Theorem 13.5.3 Suppose ao,..., an G C, n G N, an ^ 0,

n

0

Then, P(z) = 0 for some z € C.

Proof. Without loss of generality, we may assume that an = 1.
Put n = inf26C \P(z)- If \A = R then

\P{z)\ > Rn(l - {a^R-^1 \a 0 \ R~n).