540 Chapter 8 n Infinite Series of Real Numbers
Definition 8.8.16 7f = 2u, where u = min{t > 0: C(t) = O}.
(That is, ~ is the smallest positive real number x such that C(x) = 0.)
Theorem 8.8.17 (a) CG)= 0, S(~) = l.
(b) S(7r) = O; C(7r) = -1; S(27r) = O; C(27r) = l.
(c) Vx E JR, S(~ - x) = C(x), and cm -x) = S(x).
(d) S(x) is increasing on[-~,~] and C(x) is decreasing on [O, ~].Theorem 8.8.18 S(x) and C(x) are periodic with period 27r. That is, 27f is the
smallest real number k such that Vx E JR, S(x+k) = S(x), and C(x+k) = C(x).
Proof. First show that when k = 27r, Vx E JR, S(x + k) = S(x) and
C(x + k) = C(x). Then, for contradiction, suppose :3k 3 0 < k < 27f satisfying
these equations. Show that S(k) = 0 and C(k) = 1, and then using Theorem
8.8.13, show that C( ~) = l. This would contradict (8.8.15) and (8.8.16). Finally,
show that Vx E JR, S(x + k) = S(x)? Vx E JR, C(x + k) = C(x). •
Theorem 8.8.19 (a) The graph of S(x) is symmetric relative to the line
x = ~; that is, Vx E JR, S(~ - x) = S(~ + x);
(b) s ( x) is decreasing on rn'^3 ;J and increasing on [^3 ; ) 27f].
(c) C(x) is decreasing on [O, 7r] and increasing on [7r, 27r].Theorem 8.8.20 The functions S(x) and C(x) are identical to the functions
sinx and cosx defined in Definitions 7. 7.22 and 7. 7.29.
Proof. See Theorems 7.7.34 and 7.7.35. •Having defined the functions S(x) = sinx and C(x) = cosx, we define the
remaining trigonometric functions in the usual way, as given in Table 6 .1 in
Section 6.2.