Yet Another Calculus Text, 2007a

16 CHAPTER 1. DERIVATIVES
y
1
x 2 + y 2 =1
0.75
B =(cos(t), sin(t))
0.5
t
0.25
0
C
A
−1 −0.75 −0.5 −0.25 0 0.25 0.5 0.75 1
x
Figure 1.5.1: An arc of length t on the unit circle
for all values of t. Equivalently,
Solving for cos(t), we may also write this as
0 ≤ 1 − cos(t) ≤ 1 2 t2 (1.5.13)
1 − t2 2
≤ cos(t) ≤ 1 (1.5.14)
for all t.
In particular, if ɛ is an infinitesimal, then
1 − ɛ2 2
≤ cos(ɛ) ≤ 1 (1.5.15)
implies that
cos(0 + ɛ) =cos(ɛ) ≃ 1=cos(0). (1.5.16)
That is, the function f(t) =cos(t) is continuous at t =0.
Moreover, since 0 ≤ 1+cos(t) ≤ 2 for all t,
sin 2 (t) =1− cos 2 (t)
=(1− cos(t))(1 + cos(t))
≤ t2 (1 + cos(t))
2
≤ t 2 , (1.5.17)
from which it follows that
| sin(t)| ≤|t| (1.5.18)