Geometry with Trigonometry

144 Trigonometry; cosine and sine; addition formulae Ch. 9

We suppose first that|α|◦=|β|◦≤90 so thatP∈Q 1 ,P′∈Q′ 1 .ThenU∈
[O,Q],V∈[O,R],U′∈[O′,Q′],V′∈[O′,R′]. The triangles[O,U,P],[O′,U′,P′]
are congruent by the ASA-principle, so|O,U|=|O′,U′|,|O,V|=|O′,V′|.Then
|Q,U|=|Q′,U′|,|R,V|=|R′,V′|. Hence cosα=cosβ,sinα=sinβ.
Similar arguments work in the case of the other three quadrants ofF.
Conversely, let cosα=cosβ,sinα=sinβ. Suppose first that cosα≥ 0 ,sinα≥

0. ThenP∈Q 1 ,P′∈Q′ 1 .But|Q,U|=|Q′,U′|,|R,V|=|R′,V′|and so|O,U|=
   |O′,U′|,|U,P|=|U′,P′|. By the SSS-principle, the triangles[O,U,P],[O′,U′,P′]
   are congruent so|α|◦=|β|◦, unless we have a degeneration from a triangle and one
   angle is null and the other is full.
   A similar argument works for the other three quadrants ofF.

Exercises

9.1 Prove that for all anglesα∈A∗(F),

− 1 ≤cosα≤ 1 ,− 1 ≤sinα≤ 1.

9.2 LetC 1 be the circle with centre O and radius of lengthk.LetZ 1 ≡

(kcosθ,ksinθ), Z 2 ≡(−k, 0 ),Z 3 ≡(k, 0 ),sothatZ 1 is a point on this cir-

cle, and[Z 2 ,Z 3 ]is a diameter. LetC 2 be the circle with[Z 1 ,Z 3 ]as diameter.

Find the coordinates of the second point in whichC 2 meets the lineZ 2 Z 3 .How

does this relate to 4.3.3?

9.3 IfDis the mid-point of the side[B,C]of the triangle[A,B,C]andd 1 =|A,D|,

prove that

4 d 12 =b^2 +c^2 + 2 bccosα.

Deduce that 2d 1 >aif and only ifαis an acute angle.

9.4 Prove the identities

cosα+cosβ=2cos(^12 α+^12 β)cos(^12 α−^12 β),

cosα−cosβ=−2sin(^12 α+^12 β)sin(^12 α−^12 β),

and find similar results for sinα+sinβand sinα−sinβ.

9.5 Show that

sin270F+sin210F=−

3

2

,

and yet

2sin[^12 ( (^270) F+ (^210) F)]cos[^12 ( (^270) F− (^210) F)] =

3

2

.