140 Trigonometry; cosine and sine; addition formulae Ch. 9
Ifα= (^180) Fthen cosδ=0, so that sinδ=±1andsoδis either 90For 270F.
COMMENT. Our definition of a half-angle is the standard one for the angles we
deal with, but it would not suit angles which we do not consider, for example ones
with degree-magnitude greater than 360. The latter are difficult to give an account of
geometrically. For us^12 α+^12 βand^12 (α+β)need not be equal; we shall deal with
such matters in 12.1.1. Because of this, there is a danger of error if half-angles are
used incautiously.
For any anglesα,β∈A(F),ifγ=^12 α+^12 βandδ=^12 α−^12 β,thenγ+δ=α
andγ−δ=β.
Proof. As we are dealing with a commutative group, we have
γ+δ=
[ 1
2 α+1
2 β]
+
[ 1
2 α+(
−^12 β)]
=
[ 1
2 α+1
2 α]
+
[ 1
2 β+(
−^12 β)]
=α+ (^0) F=α.
Similarly
γ−δ=
[ 1
2 α+1
2 β]
−
[ 1
2 α+(
−^12 β)]
=
[ 1
2 α+1
2 β]
+
[(
−^12 α)
+^12 β]
=β.9.5 Thecosineandsinerules .......................
9.5.1 Thecosinerule.............................
NOTATION.LetA,B,Cbe non-collinear points. Then for the triangle[A,B,C]we
denote byathe length of the side which is opposite the vertexA,bybthe length of
the side oppositeB, and bycthe length of the side oppositeC,sothat
a=|B,C|,b=|C,A|,c=|A,B|.We also use the notation
α=∠BAC,β=∠CBA,γ=∠ACB.Let A,B,C be non-collinear points, let D=πBC(A)and write x=|B,D|.Then
with the notation above, 2 ax=a^2 +c^2 −b^2 when D∈[B,C]or C∈[B,D], while
2 ax=b^2 −a^2 −c^2 when B∈[D,C].
Proof.WhenD∈[B,C]we have|D,C|=a−x,andwhenC∈[B,D],|D,C|=
x−a. In both of these cases, by Pythagoras’ theorem used twice we have
|A,D|^2 =|A,B|^2 −|B,D|^2 =c^2 −x^2 ,|A,D|^2 =|A,C|^2 −|D,C|^2 =b^2 −(a−x)^2.On equating these we havec^2 −x^2 =b^2 −a^2 + 2 ax−x^2 ,giving2ax=c^2 +a^2 −b^2.
WhenB∈[D,C]we have|D,C|=a+x, so by the formulae for|A,D|^2 above we
havec^2 −x=b^2 −(a+x)^2. This simplifies to 2ax=b^2 −a^2 −c^2.