Geometry with Trigonometry

Sec. 12.5 Extension of domains of cosine and sine 249

so for some complex constantc,

E(x+y)=cE(x),for allx∈R.

On puttingx=0 we see thatc=E(y)and soE(x+y)=E(x)E(y)for allx,y∈R.

FromE′(x)=ıE(x)∀x∈R, we deduce thatC′(x)=−S(x),S′(x)=C(x),∀x∈R.

From these the usual Maclaurin series forC(x)andS(x)(orE(x)) can be derived and

tables of approximations to them prepared; see e.g. Apostol [1, Volume I, pages 435–

436]. By introducing arctanx,πcould be expressed as the sum of infinite series and

these used to prepare approximations efficiently; see e.g. Apostol [1, Volume I, pages

253–255, 284–285].

12.5.2 .....................................

Given a frame of referenceF,letQ≡( 1 , 0 )and consider the pointsZof the unit
circleC(O;1)and the anglesθ=∠FQOZinA(F).Ifs=|θ|rthenZ∼cisθ=
C(s)+ıS(s). Any numbert∈Rsuch that

C(t)+ıS(t)=C(s)+ıS(s)

is called a radian measure ofθ; collectively we shall call them themultiple radian

measuresofθ. They have the formt=s+ 2 nπıwheren∈Z. We writet∼∠FQOZ.

Suppose now thatt 1 ∼∠FQOZ 1 =θ 1 andt 2 ∼∠FQOZ 2 =θ 2 .Then

E(t 1 )=C(t 1 )+ıS(t 1 )=cosθ 1 +ısinθ 1 ,

E(t 2 )=C(t 2 )+ıS(t 2 )=cosθ 2 +ısinθ 2 ,

and from this by multiplication we have that

E(t 1 +t 2 )=E(t 1 )E(t 2 )=cos(θ 1 +θ 2 )+ısin(θ 1 +θ 2 ).

Thust 1 +t 2 ∼∠FQOZ 1 +∠FQOZ 2 and so addition of multiple measures corre-

sponds to addition of angles.

Given anyt∈R,thenγ(u)=E(ut)( 0 ≤u≤ 1 )gives a curve on the unit circle,

with initial pointQand terminal pointZ.Ift∈[− 2 π, 2 π],γwinds around the origin

and the curves generated in this way give us some pictorial feel for these multiple

radian measures of angles.