Geometry with Trigonometry

140 Trigonometry; cosine and sine; addition formulae Ch. 9 Ifα= (^180) Fthen cosδ=0, so that sinδ=±1andsoδis either 90For 270F. ...

Sec. 9.5 The cosine and sine rules 141 A B C D Figure 9.12. A B C D A B D C The cosine rule.In each triangle[A,B,C], cos ...

142 Trigonometry; cosine and sine; addition formulae Ch. 9 so that sin^2 α a^2 = 4 b^2 c^2 −(b^2 +c^2 −a^2 )^2 4 a^2 b^2 c^2 = 2 ...

Sec. 9.6 Cosine and sine of angles equal in magnitude 143 Proof. By the last result we haved 2 =c^2 +caacos^12 β,d 3 =a^2 +abbco ...

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] ...

10 Complex coordinates; sensed angles; angles between lines COMMENT. In this chapter we utilise complex coordinates, develop sen ...

146 Complex coordinates; sensed angles; angles between lines Ch. 10 (ii)If Z 1 =Z 2 ,thenZ∈Z 1 Z 2 if and only if z−z 1 =t(z 2 ...

Sec. 10.1 Complex coordinates 147 from which we havey 4 −y 3 =t(y 2 −y 1 ). Whenx 2 −x 1 =0, by (10.1.1) we must havex 4 −x 3 =0 ...

148 Complex coordinates; sensed angles; angles between lines Ch. 10 10.2 Complex-valueddistance ....................... 10.2.1Co ...

Sec. 10.2 Complex-valued distance 149 O I J H^1 H 2 H 4 H 3 Z 0 I 0 J 0 Z θ Figure 10.1. O I J H^1 H 2 H 4 H 3 Z 0 I ...

150 Complex coordinates; sensed angles; angles between lines Ch. 10 (iii) For by (i) and (ii) of the present theorem, cisθ.cis(− ...

Sec. 10.3 Rotations and axial symmetries 151 which has the matrix form ( x′−x 0 y′−y 0 ) = ( cosα −sinα sinα cosα )( x−x 0 y−y 0 ...

152 Complex coordinates; sensed angles; angles between lines Ch. 10 Let l be the line Z 0 Z 1 ,Z 0 ∼Fz,F′=tO,Z 0 (F),I 0 =tO,Z 0 ...

Sec. 10.4 Sensed angles 153 and soz′−z 0 =( ̄z−z ̄ 0 )cis 2α. Hence x′−x 0 +ı(y′−y 0 )=[x−x 0 −ı(y−y 0 )](cos2α+ısin2α), so that ...

154 Complex coordinates; sensed angles; angles between lines Ch. 10 (iii) If the points Z 1 and Z 2 are both distinct from Z 0 , ...

Sec. 10.4 Sensed angles 155 (ii) From (i) ℑ z 2 −z 0 z 1 −z 0 = |z 2 −z 0 | |z 1 −z 0 | sin(θ 2 −θ 1 ), and this is positive or ...

156 Complex coordinates; sensed angles; angles between lines Ch. 10 10.5 Sensed-area .............................. 10.5.1 ..... ...

Sec. 10.5 Sensed-area 157 Note that δF(Z 1 ,Z 2 ,Z 3 )=δF(Z 2 ,Z 3 ,Z 1 )=δF(Z 3 ,Z 1 ,Z 2 ) =−δF(Z 1 ,Z 3 ,Z 2 )=−δF(Z 2 ,Z 1 , ...

158 Complex coordinates; sensed angles; angles between lines Ch. 10 For the left-hand side here is equal to 1 2 det ⎛ ⎝ x 4 y 4 ...

Sec. 10.6 Isometries as compositions 159 LetF=([O,I,[O,J)andF 1 =([Z 0 ,Z 1 ,[Z 0 ,Z 2 )be frames of reference. Let tO,Z 0 (I)=I ...