Geometry with Trigonometry

Sec. 12.4 Parametric equations for a circle 247

12.4.2Lengthofanarcofacircle ......................

For P∈C(O;k)letαbe an angle inA∗(F)with support|QOP.Let|α|◦=θandγ
be the curve with parametric equations x=kc(t),y=ks(t)( 0 ≤t≤θ).Thenγhas
the end-points Q and P and has length 180 πθk.
Proof. The length ofγis given by
∫θ

0

√

[kc′(t)]^2 +[ks′(t)]^2 dt=

π

180

θk.

NOTE. In the foregoing result, whenθ=360,γis the circleC(O;k)and has
length 2πk.Whenθ=180,γis a semi-circle and has lengthπk. For other cases, we
note that the lineQPhas equation

s(θ)(x−k)−[c(θ)− 1 ]y= 0.

The left-hand side in this has the value−ks(θ)whenx=y=0, and whenx=
kc(t),y=ks(t)the value

k{s(θ)[c(t)− 1 ]−[c(θ)− 1 ]s(t)}=k[− 2 s(θ)s^2 (^12 t)+ 2 s^2 (^12 θ)s(t)]

= 4 ks(^12 θ)s(^12 t)

[

s(^12 θ)c(^12 t)−s(^12 t)c(^12 θ)

]

= 4 ks(^12 θ)s(^12 t)s(^12 θ−^12 t)

> 0 ,for 0<t<θ.

Whenθ<180, we have−s(θ)<0sothatZ≡(kc(t),ks(t))is in the closed half-
planeH 6 with edgeQPwhich does not containO. Thus, recalling 7.5,γis the minor
arc with end-pointsQandP.
When 180<θ< 360 ,−s(θ)>0soZis in the closed half-planeH 5 with edge
QPwhich containsO. Thusγis the major arc with end-pointsQandP.

12.4.3 Radian measure ...........................

Definition. If, for any angleα,x=|α|◦then 180 πxis called theradian measureof
α, and denoted by|α|r. We also define, for 0≤x≤ 2 π,

C(x)=c

(

180

π

x

)

,S(x)=s

(

180

π

x

)

,

andthenbythechainrulehave

C′(x)=−S′(x),S′(x)=C(x),

for 0<x< 2 π.