# Geometry with Trigonometry

## Description

## Key Features

- 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

## Readership

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

## Table of Contents

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

## Product details

- No. of pages: 280
- Language: English
- Copyright: © Woodhead Publishing 2015
- Published: November 26, 2015
- Imprint: Woodhead Publishing
- Hardcover ISBN: 9780128050668
- eBook ISBN: 9780128050675

## About the Author

### Patrick Barry

*Patrick D. Barry*, National University of Ireland, Ireland