Topics: Trigonometry, Trigonometric functions, Law of cosines Pages: 45 (13827 words) Published: April 7, 2013
CONTENTS
1.0 Introduction to Trigonometry3
2.0 Origin of Trigonometry4
2.1 Etymology4
2.2 Early Trigonometry5
2.3 Greek Findings on Trigonometry5
2.4 Indian Findings on Trigonometry7
2.5 Islamic Findings on Trigonometry16
2.6 Chinese Findings on Trigonometry18
2.7 Further European Findings on Trigonometry.19
3.0 Trigonometric Functions20
3.1 Right – Angled Triangle Definitions21
3.2 Sine, Cosine and Tangent23
3.3 Inverse Functions24
3.4 Reciprocal Functions25
3.5 Slope Definitions26
3.6 Unit-Circle Definitions26
3.7 Series Definitions31
3.8 Relationship To Exponential Function And Complex Numbers.33 3.9 Complex Graphs34
4.0 Trigonometric Identities36
4.1 Notation36
4.2 Pythagorean Identity37
4.2.1 Related Identities37
4.3 Historic Shorthand38
4.4 Symmetry, Shifts and Periodicity40
4.4.1 Symmetry40
4.4.2 Shifts and periodicity40
4.5 Angle Sum and Difference Identities41
4.5.1 Matrix Form42
4.5.2 Sines and Cosines of Sums of Infinitely Many Terms42
4.5.3 Tangents of Sums43
4.5.4 Secants and Cosecants of Sums44
4.6 Properties and Applications45
4.6.1 Law of Sines45
4.6.2 Law of cosines45
4.6.2 Law of Tangents46
4.6.3 Law of cotangents46
5.0 Uses Of Trigonometry48
5.1 Thomas Paine's Statement of The Uses of Trigonometry48
5.2 Some Modern Uses48
5.2.1 Fourier series49
5.2.2 Fourier transforms50
5.2.3 Statistics, including mathematical psychology50
5.2.4 A simple experiment with polarized sunglasses51
5.2.4 Number Theory51
5.2.5 Solving Non-Trigonometric Equations52
5.3 Trigonometry in Science and Engineering52
5.3.1 Physics52
5.3.2 Astronomy52
5.3.3 Chemistry52
5.3.4 Geography, Geodesy, and Land Surveying52
5.3.6 Engineering53
6.0 Summary54

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1.0 Introduction to Trigonometry

T
rigonometry (from Greek trigōnon "triangle" + metron "measure") is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves. The field evolved during the third century BC as a branch of geometry used extensively for astronomical studies. It is also the foundation of the practical art of surveying. The Canadarm2 robotic manipulator on the International Space Station is operated by controlling the angles of its joints. Calculating the final position of the astronaut at the end of the arm requires repeated use of trigonometric functions of those angles. The Canadarm2 robotic manipulator on the International Space Station is operated by controlling the angles of its joints. Calculating the final position of the astronaut at the end of the arm requires repeated use of trigonometric functions of those angles. Trigonometry basics are often taught in school either as a separate course or as part of a pre-calculus course. The trigonometric functions are pervasive in parts of pure mathematics and applied mathematics such as Fourier analysis and the wave equation, which are in turn essential to many branches of science and technology. Trigonometry studies triangles on spheres, surfaces of constant positive curvature, in elliptic geometry. It is fundamental to astronomy and navigation. Trigonometry on surfaces of negative curvature is part of hyperbolic geometry.

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2.0 Origin of Trigonometry

T
rigonometry is a field of mathematics first compiled in 2nd century BC by the Greek mathematician Hipparchus. The history of trigonometry and of trigonometric functions follows the general lines of the history of mathematics. In 1595, the mathematician Bartholemaeus Pitiscus published an influential work on trigonometry in 1595 which may have coined the word "trigonometry". Early study of triangles can be traced...