BASIC PRINCIPLES OF MATHEMATICS

TABLE OF CONTENTS

INTRODUCTION ..... 1

Chapter 1

Plane Trigonometry

1.1 DEFINITION ..... 2

1.2 TYPES OF TRIANGLES ..... 2

1.2.1 Arbitrary triangles ..... 2
1.2.2 Right angled triangles ..... 2
1.2.3 Spherical triangles ..... 3
1.2.4 Types of angles ..... 3
1.2.5 Measurements ..... 3

1.3 GONIOMETRIC CIRCLE ..... 4

1.3.1 Arcs ..... 4

A. Complementary Arcs ..... 4
B. Supplementary Arcs ..... 4

1.3.2 Angle a ..... 5

1.4 SINE AND COSINE ..... 7

1.4.1 The unity ..... 8
1.4.2 Signes ..... 11

1.4.2.1 Sine ..... 11
1.4.2.2 Goniometric relations of the sine ..... 12
1.4.2.3 Cosine ..... 12
1.4.2.4 Goniometric relations of the cosine ..... 13
1.4.2.5 Use of Sine and Cosine ..... 14
1.4.2.6 Fundamental Identities ..... 14

1.5 TANGENT AND COTANGENT ..... 16

1.5.1 Signes ..... 17

1.5.1.1 Tangent ..... 18
1.5.1.2 Goniometric relations of the tangent ..... 19
1.5.1.3 Cotangent ..... 20
1.5.1.4 Goniometric relations of the cotangent ..... 20
1.5.1.5 Use of tangent and cotangent ..... 21

1.6 SECANT ..... 22

1.6.1 Signes ..... 22
1.6.2 Goniometric relations of the secant ..... 23

1.7 COSECANT ..... 24

1.7.1 Signes ..... 24
1.7.2 Goniometric relations of the cosecant ..... 25

1.8 SUMMARY ..... 26

1.9 BASIC FORMULA ..... 27 

1.10 OTHER IMPORTANT TRIGONOMETRIC FORMULAS ..... 28

1.11 TYPICAL TRIGONOMETRIC VALUES ..... 31

1.12 RIGHT ANGLED TRIANGLES ..... 32

1.12.1 Theorem I ..... 32
1.12.2 Theorem II ..... 33
1.12.3 Theorem III ..... 33
1.12.4 Theorem IV ..... 33
1.12.5 Theorem V (Pythagoras Theorem) ..... 33

A. FURTHER COMMENTS ..... 34
B. EXERCISES ..... 34
C. NOTE ..... 36
D. PRACRICAL APPLICATION FOR THE NAVIGATOR ..... 36

1.13 TRIANGLES WITH NO RIGHT ANGLES ..... 37

1.13.1 Theorem I ..... 37
1.13.2 Theorem II ..... 37

Chapter 2

Spherical Trigonometry

2.1 DEFINITION ..... 39                           

2.2 PURPOSE OF SPHERICAL TRIGONOMETRY ..... 39            

2.3 SPHERE ..... 39

2.3.1 Radius of a sphere ..... 39                               
2.3.2 Diameter of a sphere ..... 39
2.3.3 Plane section of a sphere ..... 39
2.3.4 Types of spheres ..... 40

2.4 GREAT CIRCLES AND SMALL CIRCLES ..... 41

2.4.1 Particularities relating to great circles ..... 42
2.4.2 Angular distance ..... 42
2.4.3 Angular radius ..... 43
2.4.4 Polar distance ..... 43

2.5 SPHERICAL ANGLES ..... 44

2.6 SPHERICAL TRIANGLES ..... 45

2.7 FORMULAS ..... 46

2.7.1 Right angled spherical triangles ..... 46

2.7.1.1 Cosine Rule ..... 47
2.7.1.2 Sine Rule ..... 47

2.7.2 Spherical triangles with no right angles ..... 49

2.7.2.1 Group one ..... 49
2.7.2.2 Group two ..... 49
2.7.2.3 Group three ..... 50

Chapter 3

HAVERSINES

3.1 DEFINITION ..... 5

1 3.2 VERSINE ..... 51

Chapter 4

RADIANS

4.1 INTRODUCTION ..... 53

4.2 DEFINITION ..... 53

4.3 Pi ..... 53

4.4 RADIAN ..... 54

4.5 RELATION BETWEEN DEGREES AND RADIANS ..... 54

4.6 ABOUT DEGREES ..... 55

4.6.1 Degrees ..... 55
4.6.2 Angles in trigonometry ..... 55
4.6.3 DMS degrees ..... 55
4.6.4 Decimal degrees ..... 55
4.6.5 Converting decimal degrees to DMS and DMS to decimal degrees 56

4.7 RADIANS ..... 57

4.8 USE OF RADIANS AND DEGREES ..... 57

4.8.1 Convert degrees to radians ..... 57
4.8.2 Convert radians to degrees .....57

EXERCISES ..... 58

A. Area using degrees ..... 58
B. Area using radians ..... 58
C. Lenght of arc AB using degrees ..... 59
D. Lenght of arc AB using radians ..... 59

Chapter 5

LOGARITMS

5.1 INTRODUCTION ..... 60

5.2 DEFINITION ..... 60

5.3 TYPES OF LOGARITMS ..... 60

5.4 COMMON LOGARITMS ..... 61

5.5 STRUCTURE OF LOGARITHMS ..... 6

5.5.1 Index or characteristic ..... 63

5.5.1.1 Index of whole numbers ..... 63
5.5.1.2 Index of numbers with decimal value ..... 63
5.5.1.3 Index of decimal numbers ..... 63
5.5.1.4 Negative index ..... 63
5.5.2 The mantissa ..... 64
5.5.3 Cologarithms ..... 65

5.6 USE OF THE TABLES OF LOGARITHMS ..... 66

5.6.1 Find the log corresponding to a given number ..... 66

5.6.1.1 The number consists of maximum 4 figures ..... 66
5.6.1.2 The number consists of more than 4 figures ..... 67

5.6.2 Find the number corresponding to a log ..... 67

5.6.2.1 The given logarithm is in the tables ..... 67
5.6.2.2 The given logarithm is not in the tables ..... 68

5.9 LOGARITHMS OF TRIGONOMETRICAL FUNCTIONS ..... 69

5.8 THE USE OF LOGARITHMS ..... 72

5.8.1 Worked examples ..... 72

5.8.1.1 Multiplications ..... 72
5.8.1.2 Divisions ..... 72
5.8.1.3 Powers ..... 73
5.8.1.4 Roots ..... 73
5.8.1.5 Trigonometry .....7

Right angled triangle ..... 74
Triangle with no right angle ..... 74