Geometry with Trigonometry

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 years
• Errors have been corrected and some proofs marginally improved
• The 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 given

• Provides a modern and coherent exposition of geometry with trigonometry for many audiences across mathematics
• Provides many geometric diagrams for a clear understanding of the text and includes problem exercises for many chapters
• Generalizations of this material, such as to solid euclidean geometry and conic sections, when combined with calculus, would lead to applications in science, engineering, and elsewhere

A reference for second year students in Plane Euclidean Geometry and many audiences across classical applied mathematics.

Preface

• Acknowledgements
• PREFATORY NOTE TO THE REVISED EDITION

Glossary

• Greek and Latin roots of mathematical words

1: Preliminaries

• 1.1 Historical note
• 1.2 Note on deductive reasoning
• 1.3 Euclid's the elements
• 1.4 Our approach
• 1.5 Revision of geometrical concepts
• 1.6 Pre-requisites

2: Basic shapes of geometry

• 2.1 Lines, segments and half-lines
• 2.2 Open and closed half-planes
• 2.3 Angle-supports, interior and exterior regions, angles
• 2.4 Triangles and convex quadrilaterals
• Exercises

3: Distance; degree-measure of an angle

• 3.1 Distance
• 3.2 Mid-points
• 3.3 A ratio result
• 3.4 The cross-bar theorem
• 3.5 Degree-measure of angles
• 3.6 Mid-line of an angle-support
• 3.7 Degree-measure of reflex angles
• Exercises

4: Congruence of triangles; parallel lines

• 4.1 Principles of congruence
• 4.2 Alternate angles, parallel lines
• 4.3 Properties of triangles and half-planes
• Exercises

5: The parallel axiom; Euclidean geometry

• 5.1 The parallel axiom
• 5.2 Parallelograms
• 5.3 Ratio results for triangles
• 5.4 Pythagoras' theorem, c. 550B.C.
• 5.5 Mid-lines and triangles
• 5.6 Area of triangles, and convex quadrilaterals and polygons
• Exercises

6: Cartesian coordinates; applications

• 6.1 Frame of reference, cartesian coordinates
• 6.2 Algebraic note on linear equations
• 6.3 Cartesian equation of a line
• 6.4 Parametric equations of a line
• 6.5 Perpendicularity and parallelism of lines
• 6.6 Projection and axial symmetry
• 6.7 Coordinate treatment of harmonic ranges
• Exercises

7: Circles; their basic properties

• 7.1 Intersection of a line and a circle
• 7.2 Properties of circles
• 7.3 Formula for mid-line of an angle-support
• 7.4 Polar properties of a circle
• 7.5 Angles standing on arcs of circles
• 7.6 Sensed distances
• Exercises

8: Translations; axial symmetries; isometries

• 8.1 Translations and axial symmetries
• 8.2 Isometries
• 8.3 Translation of frame of reference
• Exercises

9: Trigonometry; cosine and sine; addition formulae

• 9.1 Indicator of an angle
• 9.2 Cosine and sine of an angle
• 9.3 Angles in standard position
• 9.4 Half angles
• 9.5 The cosine and sine rules
• 9.6 Cosine and sine of angles equal in magnitude
• Exercises

10: Complex coordinates; sensed angles; angles between lines

• 10.1 Complex coordinates
• 10.2 Complex-valued distance
• 10.3 Rotations and axial symmetries
• 10.4 Sensed angles
• 10.5 Sensed-area
• 10.6 Isometries as compositions
• 10.7 Orientation of a triple of non-collinear points
• 10.8 Sensed angles of triangles, the sine rule
• 10.9 Some results on circles
• 10.10 Angles between lines
• 10.11 A case of pascal's theorem, 1640
• Exercises

11: Vector and complex-number methods

• 11.1 Equipollence
• 11.2 Sum of couples, multiplication of a couple by a scalar
• 11.3 Scalar or dot products
• 11.4 Components of a vector
• 11.5 Vector methods in geometry
• 11.6 Mobile coordinates
• 11.7 Some well-known theorems
• 11.8 Isogonal conjugates
• Exercises

12: Trigonometric functions in calculus

• 12.1 Repeated bisection of an angle
• 12.2 Circular functions
• 12.3 Derivatives of cosine and sine functions
• 12.4 Parametric equations for a circle
• 12.5 Extension of domains of cosine and sine