Geometry with Trigonometry
120 Circles; their basic properties Ch. 7 7.10 For 0<a<b, suppose thatA≡( 0 ,a),B≡( 0 ,b). Show that the circlesC(A;a) and ...
8 Translations; axial symmetries; isometries COMMENT. In this chapter we introduce translations and develop them and axial symme ...
122 Translations; axial symmetries; isometries Ch. 8 (vi)If Z 1 =Z 2 ,Z∈Z 1 Z 2 and W=tZ 1 ,Z 2 (Z),then[Z 1 ,Z 2 ,W,Z]is a pa ...
Sec. 8.2 Isometries 123 and thus |sl(Z 3 ),sl(Z 4 )|^2 = 1 (a^2 +b^2 )^2 { [(b^2 −a^2 )(x 3 −x 4 )− 2 ab(y 3 −y 4 )]^2 +[− 2 ab( ...
124 Translations; axial symmetries; isometries Ch. 8 (ix)If l and m are intersecting lines, then f(l)and f(m)are intersecting li ...
Sec. 8.2 Isometries 125 (iv) TakeZ 3 =Z 1 so thatZ 1 ∈[Z 2 ,Z 3 ].ThenZ 1 Z 2 =[Z 1 ,Z 2 ∪[Z 1 ,Z 3. Hence f(Z 1 Z 2 )=f([Z 1 , ...
126 Translations; axial symmetries; isometries Ch. 8 (viii) LetW∈H 3 .Thenby(vii)W=f(Z)for someZ∈Π.IfW∈f(l)then by (iv)Z∈l⊂H 1 . ...
Sec. 8.3 Translation of frame of reference 127 H 4 ′.Thenif(x′,y′)are the Cartesian coordinates ofZ′relative to([O′,I′,[O′,J′), ...
128 Translations; axial symmetries; isometries Ch. 8 Exercises 8.1 IfT is the set of all translations ofΠ, show that(T,◦)is a co ...
9 Trigonometry; cosine and sine; addition formulae COMMENT. In this chapter we go on to deal fully with reflex-angles as well as ...
130 Trigonometry; cosine and sine; addition formulae Ch. 9 Definition. Referring to 2.3.3, for each angle support|BACletl=ml(|BA ...
Sec. 9.2 Cosine and sine of an angle 131 To help us in our study of angles, it is convenient to fit a frame of reference to the ...
132 Trigonometry; cosine and sine; addition formulae Ch. 9 When insteadP∈H 4 ,wehaveO∈[Q,U]so|O,U|=|Q,U|−kand similarly |O,U 1 | ...
Sec. 9.3 Angles in standard position 133 Proof.LetQ,Rbe the points whereC(O;k)meets[O,I and[O,J, respectively; thenQandRhave Car ...
134 Trigonometry; cosine and sine; addition formulae Ch. 9 P O J I Q R S T i(α) H 1 H 2 H 4 H 3 P O J I Q R S T i(α) ...
Sec. 9.3 Angles in standard position 135 Ifαandγare different angles inA∗(F),then|α|◦=|γ|◦. Proof. This is evident ifαandγare b ...
136 Trigonometry; cosine and sine; addition formulae Ch. 9 We note that in 7.3.1 (a 1 +a 2 )^2 +(b 1 +b 2 )^2 = 2 ( 1 +a 1 a 2 + ...
Sec. 9.3 Angles in standard position 137 Modified addition + of angles has the following properties:- (i) For allα,β∈A(F),α+βis ...
138 Trigonometry; cosine and sine; addition formulae Ch. 9 9.3.4 Subtractionofangles ......................... Definition.Forall ...
Sec. 9.4 Half angles 139 (i) cos90F= 0 ,sin90F= 1 ,cos180F=− 1 , sin180F= 0 ,cos270F= 0 ,sin270F=− 1. (ii) 2( (^90) F)= (^180) F ...
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