Geometry with Trigonometry

248 Trigonometric functions in calculus Ch. 12

12.5 Extensionofdomainsofcosineandsine ...............

12.5.1 .....................................

NOTE.IfwedefineE(x)byE(x)=C(x)+ıS(x)( 0 ≤x≤ 2 π),whereı^2 =−1, then
by 12.2.1(i)
E(x+y)=E(x)E(y)( 0 ≤x≤ 2 π), (12.5.1)

and by 12.4.3
E′(x)=ıE(x)( 0 <x< 2 π).

It is natural to enquire if the definition ofEcan be extended from[ 0 , 2 π]in such a
way as to preserve its basic properties.
From (12.5.1) we note that in particular

E(x+π)=−E(x)( 0 ≤x≤π),

asE(π)=−1, and if we specify that this is to hold for allx∈Rthen the domain of
definition ofEbecomes extended toR.
It follows thatE(x+ 2 π)=E(x)for allx∈Randsoforalln∈Z,E(x+ 2 nπ)=
E(x). By the chain rule, for 0<x< 2 π,

E′(x+ 2 nπ)=E′(x)=ıE(x)=ıE(x+ 2 nπ).

Thus for eachn∈Z,E′(x)exists and satisfiesE′(x)=ıE(x)for 2nπ<x<( 2 n+ 2 )π.
Moreover for 0<h< 2 π,

E( 2 nπ+h)−E( 2 nπ)

h

=

E(h)− 1

h

=

C(h)− 1

h

+ı

S(h)

h

→ı(h→ 0 ),

while for− 2 π<h<0,

E( 2 nπ+h)−E( 2 nπ)

h

=

E( 2 π+h)− 1

h

=

−E(π+h)− 1

h

=−

E(π+h)−E(π)

h

→−E′(π)(h→ 0 )

=ı.

ThusE′( 2 nπ)exists for eachn∈Zand as its value there isı=ıE( 2 nπ), we see that
E′(x)=ıE(x)for allx∈R.
We note that|E(x)|=

√

C(x)^2 +S(x)^2 =1for0≤x≤ 2 πand then asEhas period
2 π,|E(x)|=1forallx. HenceE(x)=0forallx∈R.
Then for any fixedy∈R,

d

dx

E(x+y)

E(x)

=

E(x)ıE(x+y)−E(x+y)ıE(x)

E(x)^2

= 0 ,