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Fig 1 - Cartesian coordinate system with the points (5,12) marked in green, (-3,1) in red, (-1.5,-2.5) in blue and (0,0), the origin, in violet.
221:
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450:
660:{\displaystyle {\vec {x}}(t)=c_{1}e^{\lambda _{1}t}{\vec {v}}_{1}+\ldots +c_{n}e^{\lambda _{n}t}{\vec {v}}_{n}}
1189:
871:{\displaystyle {\vec {x}}(k)=c_{1}\lambda _{1}^{n}{\vec {v}}_{1}+\ldots +c_{n}\lambda _{n}k{\vec {v}}_{n}}
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with the positive x-axis intersects the unit circle at a point whose x-coordinate (red) is the
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25:
Fig. 1a - Sine and cosine of the angle θ in the unit circle of a cartesian coordinate system.
964:{\displaystyle \langle T{\vec {v}},{\vec {w}}\rangle =\langle {\vec {v}},T{\vec {w}}\rangle }
110:
for all angles between 0 and 360°. Sine and cosine of θ are real numbers between -1 and +1.
1185:
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40:
134:
Fig. 1d - Angle θ in the fourth quadrant. The sine is negative, the cosine positive.
118:
Fig. 1b - Angle θ in the second quadrant. The sine is positive, the cosine negative.
1205:
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in 1981, under the direction of Roy
Douglas. Following a research fellowship at the
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35:
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Fig. 1c - Angle θ in the third quadrant. Both sine and cosine are negative.
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21:
331:{\displaystyle {\begin{pmatrix}1&0&1\\0&1&0\end{pmatrix}}}
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39:, which is of radius 1 and centered at the origin (see Figure 1a). The
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267:{\displaystyle {\begin{pmatrix}2&4\\-1&2\end{pmatrix}}}
210:{\displaystyle {\begin{pmatrix}1&1\\-1&1\end{pmatrix}}}
385:{\displaystyle {\begin{pmatrix}0&0\\2&2\end{pmatrix}}}
1184:
in applied homotopy theory. Since 2008, he is fellow of the
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Department of
Mathematics, University of Western Ontario
1234:
Department of
Mathematics, University of Western Ontario
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436:{\displaystyle {\begin{pmatrix}0\\1\\1\end{pmatrix}}}
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1173:, he joined the Department of Mathematics at the
1105:{\displaystyle z^{m/n}=|z|^{m/n}e^{i\arg(z)m/n}}
734:{\displaystyle {\vec {x}}(k+1)=A\,{\vec {x}}(k)}
511:{\displaystyle {\vec {x}}'(t)=A\,{\vec {x}}(t)}
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1177:in 1984, where he is currently a professor.
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54:and whose y-coordinate (green) is the
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1288:at the University of Western Ontario
1252:
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1169:and a Dickson instructorship at the
1188:, and has been recognized with the
1180:From 2002 to 2016, Jardine held a
14:
1204:Jardine is known for his work on
1163:University of British Columbia
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1296:Mathematics Genealogy Project
1194:Canadian Mathematical Society
1175:University of Western Ontario
103:{\displaystyle \cos \theta }
77:{\displaystyle \sin \theta }
154:
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31:Cartesian coordinate system
1318:
1122:(born December 6, 1951 in
1137:working in the fields of
1004:{\displaystyle 3x+4x+20=}
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1005:
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1210:simplicial presheaves
1182:Canada Research Chair
1171:University of Chicago
1167:University of Toronto
1157:Jardine obtained his
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142:Cartesian Coordinates
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1258:"CV of Rick Jardine"
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58:of θ. This defines
1280:External references
1230:"Full-time Faculty"
1190:Coxeter–James Prize
797:
1286:Jardine's homepage
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1115:Jim Bryan Article
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1120:Jim Arthur Bryan
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1139:homotopy theory
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33:, consider the
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17:Sine and Cosine
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1147:number theory
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1135:mathematician
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1300:
1267:February 11,
1265:. Retrieved
1261:
1239:February 11,
1237:. Retrieved
1233:
1224:
1203:
1179:
1156:
1119:
1118:
879:
742:
668:
519:
447:
444:
393:
339:
275:
218:
161:
153:
137:
55:
51:
47:
34:
28:
158:other stuff
36:unit circle
1216:References
1124:Belleville
1161:from the
1153:Biography
1078:
959:⟩
953:→
935:→
926:⟨
920:⟩
914:→
899:→
887:⟨
857:→
838:λ
821:…
806:→
785:λ
756:→
717:→
683:→
646:→
625:λ
603:…
588:→
567:λ
533:→
494:→
462:→
246:−
189:−
98:θ
95:
72:θ
69:
1132:Canadian
468:′
1294:at the
1130:) is a
1292:345Kai
1145:, and
1128:Canada
52:cosine
1159:Ph.D.
45:angle
29:In a
1269:2018
1241:2018
1200:Work
1149:.
84:and
56:sine
1075:arg
92:cos
66:sin
41:ray
1260:.
1249:^
1232:.
1212:.
1196:.
1141:,
1126:,
996:20
1271:.
1243:.
1098:n
1094:/
1090:m
1087:)
1084:z
1081:(
1072:i
1068:e
1062:n
1058:/
1054:m
1049:|
1044:z
1040:|
1036:=
1031:n
1027:/
1023:m
1019:z
999:=
993:+
990:x
987:4
984:+
981:x
978:3
950:w
944:T
941:,
932:v
923:=
911:w
905:,
896:v
890:T
864:n
854:v
847:k
842:n
832:n
828:c
824:+
818:+
813:1
803:v
794:n
789:1
779:1
775:c
771:=
768:)
765:k
762:(
753:x
729:)
726:k
723:(
714:x
707:A
704:=
701:)
698:1
695:+
692:k
689:(
680:x
653:n
643:v
634:t
629:n
620:e
614:n
610:c
606:+
600:+
595:1
585:v
576:t
571:1
562:e
556:1
552:c
548:=
545:)
542:t
539:(
530:x
506:)
503:t
500:(
491:x
484:A
481:=
478:)
475:t
472:(
459:x
429:)
423:1
416:1
409:0
403:(
378:)
372:2
367:2
360:0
355:0
349:(
324:)
318:0
313:1
308:0
301:1
296:0
291:1
285:(
260:)
254:2
249:1
239:4
234:2
228:(
203:)
197:1
192:1
182:1
177:1
171:(
48:θ
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