Sec. 12.3 Derivatives of cosine and sine functions 245
must exist. But the formulaμ(^1 n)=μ(n^1 )shows thatμ(^1 n)→ 0 (n→∞)and so we
must havel=0. Astn−y→0 it follows thatμ(tn−y)→ 0 (n→∞)and then
nlim→∞μ(y)=nlim→∞[tnμ(^1 )]+nlim→∞μ(tn−y)
yieldsμ(y)=yμ( 1 )+0.
To conclude we takep=μ( 1 )and note that this is positive as shown above.
12.2.2 Definition ofπ.............................
Definition. We denoteμ( 360 )byπ. Then 360p=πso thatp= 360 π and so
μ(x)=
π
360
x.
12.3 Derivativesofcosineandsinefunctions................
12.3.1 .....................................
With the notation of12.1,
tlim→ 0 +
s(t)
t
=
π
180
.
Proof. We have seen above that for 0<x≤360, takingn=2 in 12.2.1(iv),
0 <s
(x
8
)
≤
π
180
x
8
≤
s(x 8 )
c(x 8 )
,
and so for 0<t<45,
0 <s(t)≤
π
180
t≤
s(t)
c(t)
.
From the first two inequalities here we infer thats(t)→ 0 (t→ 0 +)and hence
c(t)= 1 − 2 s
(t
2
) 2
→ 1 (t→ 0 +).
As
π
180
c(t)≤
s(t)
t
≤
π
180
,
the result follows.
For 0 <x< 360 the derivatives c′(x)and s′(x)exist and are given by
c′(x)=−
π
180
s(x),s′(x)=
π
180
c(x).