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E-grāmata: Geometry with Trigonometry

(National University of Ireland, Ireland)
  • Formāts: EPUB+DRM
  • Izdošanas datums: 24-Dec-2015
  • Izdevniecība: Woodhead Publishing
  • Valoda: eng
  • ISBN-13: 9780128050675
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Geometry with TrigonometryPalielināt
  • Formāts: EPUB+DRM
  • Izdošanas datums: 24-Dec-2015
  • Izdevniecība: Woodhead Publishing
  • Valoda: eng
  • ISBN-13: 9780128050675
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Geometry with Trigonometry Second Edition is a second course in plane Euclidean geometry, second in the sense that many of its basic concepts will have been dealt with at school, less precisely. It gets underway with a large section of pure geometry in Chapters 2 to 5 inclusive, in which many familiar results are efficiently proved, although the logical frame work is not traditional. In Chapter 6 there is a convenient introduction of coordinate geometry in which the only use of angles is to handle the perpendicularity or parallelism of lines. Cartesian equations and parametric equations of a line are developed and there are several applications. In Chapter 7 basic properties of circles are developed, the mid-line of an angle-support, and sensed distances. In the short Chaper 8 there is a treatment of translations, axial symmetries and more generally isometries. In Chapter 9 trigonometry is dealt with in an original way which e.g. allows concepts such as clockwise and anticlockwise to be handled in a way which is not purely visual. By the stage of Chapter 9 we have a context in which calculus can be developed. In Chapter 10 the use of complex numbers as coordinates is introduced and the great conveniences this notation allows are systematically exploited. Many and varied topics are dealt with , including sensed angles, sensed area of a triangle, angles between lines as opposed to angles between co-initial half-lines (duo-angles). In Chapter 11 various convenient methods of proving geometrical results are established, position vectors, areal coordinates, an original concept mobile coordinates. In Chapter 12 trigonometric functions in the context of calculus are treated.New to this edition: The second edition has been comprehensively revised over three yearsErrors have been corrected and some proofs marginally improvedThe substantial difference is that Chapter 11 has been significantly extended, particularly the role of mobile coordinates, and a more thorough account of the material is givenProvides a modern and coherent exposition of geometry with trigonometry for varying levels in mathematics, applied mathematics, engineering mathematics and other areas of applicationDescribes computational geometry, differential geometry, mathematical modelling, computer science, computer-aided design of systems in mechanical, structural and other engineering, and architectureProvides many geometric diagrams for a clear understanding of the text and includes problem exercises for each chapter

Recenzijas

"The book is carefully written, and proceeds at a fairly high level of sophistication Instructors who want to present a completely rigorous development of the subject and who are therefore willing to accept the pedagogical risks involved, will certainly want to look at this text." --MAA Reviews

Papildus informācija

An account at once accessible and unobtrusively rigorous, intended to consolidate traditional synthetic geometry with progression into coordinates, trigonometry, position-vectors and complex numbers
About the author xv
Preface xvii
Glossary xix
1 Preliminaries 1(20)
1.1 Historical note
1(2)
1.2 Note on deductive reasoning
3(1)
1.2.1 Definitions
3(1)
1.2.2 Proof
3(1)
1.3 Euclid's The Elements
4(5)
1.3.1
4(1)
1.3.2 Definitions
4(2)
1.3.3 Postulates and common notions
6(1)
1.3.4
7(1)
1.3.5 Congruence
7(1)
1.3.6 Quantities or magnitudes
8(1)
1.4 Our approach
9(1)
1.4.1 Type of course
9(1)
1.4.2 Need for preparation
9(1)
1.5 Revision of geometrical concepts
10(9)
1.5.1
10(1)
1.5.2 The basic shapes
10(5)
1.5.3 Distance; degree-measure of an angle
15(2)
1.5.4 Our treatment of congruence
17(1)
1.5.5 Parallel lines
18(1)
1.6 Pre-requisites
19(2)
1.6.1 Set notation
19(1)
1.6.2 Classical algebra
20(1)
1.6.3 Other algebra
20(1)
1.6.4 Distinctive property of real numbers among fields
20(1)
2 Basic shapes of geometry 21(14)
2.1 Lines, segments and half-lines
21(5)
2.1.1 Plane, points, lines
21(1)
2.1.2 Natural order on a line
22(1)
2.1.3 Reciprocal orders
23(1)
2.1.4 Segments
23(2)
2.1.5 Half-lines
25(1)
2.2 Open and closed half-planes
26(2)
2.2.1 Convex sets
26(1)
2.2.2 Open half-planes
26(1)
2.2.3 Closed half-planes
27(1)
2.3 Angle-supports, interior and exterior regions, angles
28(2)
2.3.1 Angle-supports, interior regions
28(2)
2.3.2 Exterior regions
30(1)
2.3.3 Angles
30(1)
2.4 Triangles and convex quadrilaterals
30(5)
2.4.1 Terminology
30(1)
2.4.2 Triangles
31(1)
2.4.3 Pasch's property, 1882
32(1)
2.4.4 Convex quadrilaterals
32(3)
3 Distance; degree-measure of an angle 35(14)
3.1 Distance
35(3)
3.1.1 Axiom for distance
35(1)
3.1.2 Derived properties of distance
36(2)
3.2 Mid-points
38(1)
3.2.1
38(1)
3.3 A ratio result
39(1)
3.3.1
39(1)
3.4 The cross-bar theorem
40(1)
3.4.1
40(1)
3.5 Degree-measure of angles
40(4)
3.5.1 Axiom for degree-measure
40(1)
3.5.2 Derived properties of degree-measure
41(3)
3.6 Mid-line of an angle-support
44(2)
3.6.1 Right-angles
44(1)
3.6.2 Perpendicular lines
44(1)
3.6.3 Mid-lines
45(1)
3.7 Degree-measure of reflex angles
46(3)
3.7.1
46(3)
4 Congruence of triangles; parallel lines 49(12)
4.1 Principles of congruence
49(4)
4.1.1 Congruence of triangles
49(4)
4.2 Alternate angles, parallel lines
53(2)
4.2.1 Alternate angles
53(1)
4.2.2 Parallel lines
54(1)
4.3 Properties of triangles and half-planes
55(6)
4.3.1 Side-angle relationships; the triangle inequality
55(1)
4.3.2 Properties of parallelism
56(1)
4.3.3 Dropping a perpendicular
56(1)
4.3.4 Projection and axial symmetry
57(4)
5 The parallel axiom; Euclidean geometry 61(20)
5.1 The parallel axiom
61(1)
5.1.1 Uniqueness of a parallel line
61(1)
5.2 Parallelograms
62(2)
5.2.1 Parallelograms and rectangles
62(1)
5.2.2 Sum of measures of wedge-angles of a triangle
63(1)
5.3 Ratio results for triangles
64(5)
5.3.1 Lines parallel to one side-line of a triangle
64(3)
5.3.2 Similar triangles
67(2)
5.4 Pythagoras' theorem, c. 550 B.C.
69(1)
5.4.1
69(1)
5.5 Mid-lines and triangles
70(2)
5.5.1 Harmonic ranges
70(2)
5.6 Area of triangles, and convex quadrilaterals and polygons
72(9)
5.6.1 Area of a triangle
72(3)
5.6.2 Area of a convex quadrilateral
75(1)
5.6.3 Area of a convex polygon
75(6)
6 Cartesian coordinates; applications 81(22)
6.1 Frame of reference, Cartesian coordinates
81(4)
6.1.1
81(4)
6.2 Algebraic note on linear equations
85(1)
6.2.1
85(1)
6.3 Cartesian equation of a line
86(3)
6.3.1
86(3)
6.4 Parametric equations of a line
89(3)
6.4.1
89(3)
6.5 Perpendicularity and parallelism of lines
92(1)
6.5.1
92(1)
6.6 Projection and axial symmetry
93(2)
6.6.1
93(1)
6.6.2 Formula for area of a triangle
94(1)
6.6.3 Inequalities for closed half-planes
95(1)
6.7 Coordinate treatment of harmonic ranges
95(8)
6.7.1 New parametrisation of a line
95(2)
6.7.2 Interchange of pairs of points
97(1)
6.7.3 Distances from mid-point
98(1)
6.7.4 Distances from end-point
98(1)
6.7.5 Construction for a harmonic range
99(4)
7 Circles; their basic properties 103(18)
7.1 Intersection of a line and a circle
103(2)
7.1.1
103(2)
7.2 Properties of circles
105(2)
7.2.1
105(1)
7.2.2 Equation of a circle
106(1)
7.2.3 Circle through three points
106(1)
7.3 Formula for mid-line of an angle-support
107(1)
7.3.1
107(1)
7.4 Polar properties of a circle
108(4)
7.4.1 Tangents from an exterior point
108(1)
7.4.2 The power property of a circle
109(2)
7.4.3 A harmonic range
111(1)
7.5 Angles standing on arcs of circles
112(2)
7.5.1
112(2)
7.5.2 Minor and major arcs of a circle
114(1)
7.6 Sensed distances
114(7)
7.6.1 Sensed distance
114(3)
7.6.2 Sensed products and a circle
117(1)
7.6.3 Radical axis and coaxal circles
118(3)
8 Translations; axial symmetries; isometries 121(8)
8.1 Translations and axial symmetries
121(2)
8.1.1
121(2)
8.2 Isometries
123(4)
8.2.1
123(4)
8.2.2
127(1)
8.3 Translation of frame of reference
127(2)
9 Trigonometry; cosine and sine; addition formulae 129(16)
9.1 Indicator of an angle
129(1)
9.1.1
129(1)
9.2 Cosine and sine of an angle
130(3)
9.2.1
130(2)
9.2.2 Polar coordinates
132(1)
9.2.3
133(1)
9.3 Angles in standard position
133(6)
9.3.1 Angles in standard position
133(2)
9.3.2 Addition of angles
135(1)
9.3.3 Modified addition of angles
136(2)
9.3.4 Subtraction of angles
138(1)
9.3.5 Integer multiples of an angle
138(1)
9.3.6 Standard multiples of a right-angle
138(1)
9.4 Half angles
139(1)
9.4.1
139(1)
9.5 The cosine and sine rules
140(3)
9.5.1 The cosine rule
140(1)
9.5.2 The sine rule
141(1)
9.5.3
142(1)
9.5.4 The Steiner-Lehmus theorem, 1842
142(1)
9.6 Cosine and sine of angles equal in magnitude
143(2)
9.6.1
143(2)
10 Complex coordinates; sensed angles; angles between lines 145(40)
10.1 Complex coordinates
145(3)
10.1.1
145(3)
10.2 Complex-valued distance
148(2)
10.2.1 Complex-valued distance
148(1)
10.2.2 A complex-valued trigonometric function
148(2)
10.3 Rotations and axial symmetries
150(3)
10.3.1 Rotations
150(1)
10.3.2 Formula for an axial symmetry
151(2)
10.4 Sensed angles
153(3)
10.4.1
153(3)
10.5 Sensed-area
156(2)
10.5.1
156(1)
10.5.2 Sensed-area of a triangle
156(1)
10.5.3 A basic feature of sensed-area
157(1)
10.5.4 An identity for sensed-area
157(1)
10.6 Isometries as compositions
158(2)
10.6.1
158(2)
10.7 Orientation of a triple of non-collinear points
160(2)
10.7.1
160(2)
10.8 Sensed angles of triangles, the sine rule
162(2)
10.8.1
162(2)
10.9 Some results on circles
164(4)
10.9.1 A necessary condition to lie on a circle
164(1)
10.9.2 A sufficient condition to lie on a circle
165(1)
10.9.3 Complex cross-ratio
165(1)
10.9.4 Ptolemy's theorem, c. 200 A.D.
166(2)
10.10 Angles between lines
168(8)
10.10.1 Motivation
168(1)
10.10.2 Duo-sectors
168(1)
10.10.3 Duo-angles
169(1)
10.10.4 Duo-angles in standard position
169(2)
10.10.5 Addition of duo-angles in standard position
171(2)
10.10.6 Addition formulae for tangents of duo-angles
173(1)
10.10.7 Associativity of addition of duo-angles
174(1)
10.10.8 Group properties of duo-angles; sensed duo-angles
175(1)
10.10.9 An application
176(1)
10.11 A case of Pascal's theorem, 1640
176(9)
10.11.1
176(2)
10.11.2
178(2)
10.11.3
180(5)
11 Vector and complex-number methods 185(54)
11.1 Equipollence
185(2)
11.1.1
185(2)
11.2 Sum of couples, multiplication of a couple by a scalar
187(3)
11.2.1
187(1)
11.2.2 Vector space over R
188(2)
11.3 Scalar or dot products
190(2)
11.3.1
190(2)
11.4 Components of a vector
192(3)
11.4.1 Components
192(1)
11.4.2 Areal coordinates
193(1)
11.4.3 Cartesian coordinates from areal coordinates
193(1)
11.4.4
193(1)
11.4.5
194(1)
11.5 Vector methods in geometry
195(11)
11.5.1 Menelaus' theorem, c. 100 A.D
195(1)
11.5.2 Ceva's theorem and converse, 1678 A.D.
196(3)
11.5.3 Desargues' perspective theorem, 1648 A.D.
199(2)
11.5.4 Pappus' theorem, c. 300 A.D.
201(2)
11.5.5 Centroid of a triangle
203(1)
11.5.6 Orthocentre of a triangle
203(2)
11.5.7 Incentre of a triangle
205(1)
11.6 Mobile coordinates
206(20)
11.6.1 Grassmann's supplement of a vector
206(1)
11.6.2
207(1)
11.6.3 Handling a triangle
207(2)
11.6.4 Circumcentre of a triangle
209(2)
11.6.5 Other distinguished points for a triangle
211(4)
11.6.6 Euler line of a triangle
215(1)
11.6.7 Similar triangles
215(2)
11.6.8 Centroids of similar triangles erected on the sides of a triangle
217(1)
11.6.9 Circumcentres of similar triangles on sides of triangle
218(1)
11.6.10 Triangle with vertices the mid-points of sides of given triangle
219(3)
11.6.11 The nine-point circle
222(3)
11.6.12 Parametric equations of lines
225(1)
11.7 Some well-known theorems
226(8)
11.7.1 Feuerbach's theorem, 1822
226(3)
11.7.2 The Wallace-Simson line, 1797
229(2)
11.7.3 The incentre on the Euler line of a triangle
231(1)
11.7.4 Miquel's theorem, 1838
232(2)
11.8 Isogonal conjugates
234(5)
11.8.1 Isogonal conjugates
234(1)
11.8.2 Concurrency
235(1)
11.8.3 Symmedians
235(4)
12 Trigonometric functions in calculus 239(12)
12.1 Repeated bisection of an angle
239(2)
12.1.1
239(2)
12.2 Circular functions
241(4)
12.2.1
241(4)
12.2.2 Definition of r
245(1)
12.3 Derivatives of cosine and sine functions
245(1)
12.3.1
245(1)
12.4 Parametric equations for a circle
246(2)
12.4.1 Area of a disk
246(1)
12.4.2 Length of an arc of a circle
247(1)
12.4.3 Radian measure
247(1)
12.5 Extension of domains of cosine and sine
248(3)
12.5.1
248(1)
12.5.2
249(2)
Appendix; List of axioms 251(2)
Bibliography 253(1)
Index 254
Patrick D. Barry, National University of Ireland, Ireland
Biežāk uzdotie jautājumi par e-grāmatām Biežāk uzdotie jautājumi par e-grāmatām
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