The Way Trigonometry is used in Astronomy
By: Joanna Matthews
Practical Applications of Advanced Mathematics
Mrs. Amy Goodrum
July 15, 2003
Abstract
This report is about how trigonometry is used in Astronomy. Even though trigonometry is applied in many areas, such as engineering, chemistry, surveying, and physics, it is mainly used in astronomy Trigonometry is used to find the distance of stars, the distance from one planet to another and from one plant to the sun. It is possible to find the radius of the Earth also. This report will basically give more insight in the way trigonometry and astronomy goes hand in hand. Background
Trigonometry comes from a Greek word "trigonometria" put together from these 3 words: Tri (three) gonia (angle) metro (measure). Trigonometry has been around for many centuries, but in 140 BC a man named Hipparchus apparently wrote 12 books on the table of chords and became the founder of trigonometry. He was the first Greek mathematician to study triangular geometry. This study led him to write the 12 books. Other people have added to Hipparchus’ work, but the two people that stands out the most are were Menelaus (ca. AD 100) and Ptolemy (ca. AD 100). Menelaus was a Greek mathematician that created six books on tables of chords. He created a couple of triangle properties. Menelaus had a big hand in spherical trigonometry also. He was like the one that was after Hipparchus’ work the most. Ptolemy was a Greek astronomer who was highly respected in his city because of his work. He was the first mathematician to complete the tables of chords, which were 13 books. Although his work had respect, there was controversy behind it. People said that he stole ideas and inventions to further his work. There was no proof of these accusations and his is still respected and appreciated. The Muslims, Chinese, Indians, and Babylonians had their own information that aided to trigonometry. The Muslims introduced the tangent...
...The ancient Nubians used a similar methodology.[5] The ancient Greeks transformed trigonometry into an ordered science.[6]
Classical Greek mathematicians (such as Euclid and Archimedes) studied the properties of chords and inscribed angles in circles, and proved theorems that are equivalent to modern trigonometric formulae, although they presented them geometrically rather than algebraically. Claudius Ptolemy expanded upon Hipparchus' Chords in a Circle in his Almagest.[7] The modern sine function was first defined in the Surya Siddhanta, and its properties were further documented by the 5th century Indian mathematician and astronomer Aryabhata.[8] These Greek and Indian works were translated and expanded by medieval Islamic mathematicians. By the 10th century, Islamic mathematicians were using all six trigonometric functions, had tabulated their values, and were applying them to problems in spherical geometry.[citation needed] At about the same time, Chinese mathematicians developed trigonometry independently, although it was not a major field of study for them. Knowledge of trigonometric functions and methods reached Europe via Latin translations of the works of Persian and Arabic astronomers such as Al Battani and Nasir alDin alTusi.[9] One of the earliest works on trigonometry by a European mathematician is De Triangulis by the 15th century German mathematician Regiomontanus. Trigonometry was still so little...
...Trigonometry is a field of mathematics first compiled by 2nd century BCE. Greek mathematician Hipparchus. The history of trigonometry and of trigonometric functions follows the general lines of the history of mathematics.
Early study of triangles can be traced to the 2nd millennium BC, in Egyptian mathematics (Rhind Mathematical Papyrus) and Babylonian mathematics. Systematic study of trigonometric functions begins in Hellenistic mathematics, reaching India as part of Hellenistic astronomy. In Indian astronomy, the study of trigonometric functions flowers in the Gupta period, especially due to Aryabhata (6th century). During the Middle Ages, the study of trigonometry is continued in Islamic mathematics, whence it is adopted as a separate subject in the Latin West beginning in the Renaissance with Regiomontanus. The development of modern trigonometry then takes place in the western Age of Enlightenment, beginning with 17th century mathematics (Isaac Newton, James Stirling) and reaching its modern form with Leonhard Euler. The ancient Egyptians and Babylonians had known of theorems on the ratios of the sides of similar triangles for many centuries. But preHellenic societies lacked the concept of an angle measure and consequently, the sides of triangles were studied instead, a field that would be better called "trilaterometry".
The Babylonian astronomers kept detailed records on the rising...
...Early trigonometry
The ancient Egyptians and Babylonians had known of theorems on the ratios of the sides of similar triangles for many centuries. But preHellenic societies lacked the concept of an angle measure and consequently, the sides of triangles were studied instead, a field that would be better called "trilaterometry".[6]The Babylonian astronomers kept detailed records on the rising and setting of stars, the motion of the planets, and the solar and lunar eclipses, all of which required familiarity with angular distances measured on the celestial sphere.[2] Based on one interpretation of the Plimpton 322 cuneiform tablet (c. 1900 BC), some have even asserted that the ancient Babylonians had a table of secants.[7] There is, however, much debate as to whether it is a table of Pythagorean triples, a solution of quadratic equations, or a trigonometric table.The Egyptians, on the other hand, used a primitive form of trigonometry for building pyramids in the 2nd millennium BC.[2] The Rhind Mathematical Papyrus, written by the Egyptian scribe Ahmes (c. 1680–1620 BC), contains the following problem related to trigonometry:[2]"If a pyramid is 250 cubits high and the side of its base 360 cubits long, what is its seked?"Ahmes' solution to the problem is the ratio of half the side of the base of the pyramid to its height, or the runtorise ratio of its face. In other words, the quantity he found for the seked is the cotangent of the...
...CONTENTS
1.0 Introduction to Trigonometry 3
2.0 Origin of Trigonometry 4
2.1 Etymology 4
2.2 Early Trigonometry 5
2.3 Greek Findings on Trigonometry 5
2.4 Indian Findings on Trigonometry 7
2.5 Islamic Findings on Trigonometry 16
2.6 Chinese Findings on Trigonometry 18
2.7 Further European Findings on Trigonometry. 19
3.0 Trigonometric Functions 20
3.1 Right – Angled Triangle Definitions 21
3.2 Sine, Cosine and Tangent 23
3.3 Inverse Functions 24
3.4 Reciprocal Functions 25
3.5 Slope Definitions 26
3.6 UnitCircle Definitions 26
3.7 Series Definitions 31
3.8 Relationship To Exponential Function And Complex Numbers. 33
3.9 Complex Graphs 34
4.0 Trigonometric Identities 36
4.1 Notation 36
4.2 Pythagorean Identity 37
4.2.1 Related Identities 37
4.3 Historic Shorthand 38
4.4 Symmetry, Shifts and Periodicity 40
4.4.1 Symmetry 40
4.4.2 Shifts and periodicity 40
4.5 Angle Sum and Difference Identities 41
4.5.1 Matrix Form 42
4.5.2 Sines and Cosines of Sums of Infinitely Many Terms 42
4.5.3 Tangents of Sums 43
4.5.4 Secants and Cosecants of Sums 44
4.6 Properties and Applications 45
4.6.1 Law of Sines 45
4.6.2 Law of cosines 45
4.6.2 Law of Tangents 46
4.6.3 Law of cotangents 46
5.0 Uses Of Trigonometry 48
5.1 Thomas Paine's Statement of The Uses of Trigonometry 48
5.2 Some Modern Uses 48...
...Teaching trigonometry using Empirical Modelling
0303417
Abstract
The trigonometric functions sin(x), cos(x) and tan(x) are relationships that exist between the angles
and length of sides in a rightangled triangle. In Empirical Modelling terms, the angles in a triangle
and the length of the sides are observables, and the functions that connect them are the definitions.
These welldefined geometric relationships can be useful when teaching GCSElevel students about
the functions, as they provide a way to visualise what can be thought of as fairly abstract functions.
This paper looks at how different learning styles apply to Empirical Modelling, and presents a practical example of their use in a model to teach trigonometry.
1 Introduction
The trigonometric functions sin(x), cos(x) and tan(x)
are relationships that exist between the angles and
length of sides in a rightangled triangle. In Empirical Modelling terms, the angles in a triangle and the
length of the sides are observables, and the functions
that connect them are the definitions. These welldefined geometric relationships can be useful when
teaching GCSElevel students about the functions,
as they provide a way to visualise what can be
thought of as fairly abstract functions. Rather than
teaching students by showing them diagrams in an
instructive way (already a good way of doing it), a
constructive approach may allow students to gain a
better understanding...
...Trigonometry (from Greek trigōnon "triangle" + metron"measure"[1]) 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.[2] It is also the foundation of the practical art of surveying.
Trigonometry basics are often taught in school either as a separate course or as part of a precalculus 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. Spherical 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.


\History
Main article: History of trigonometry
The first trigonometric tablewas apparently compiled byHipparchus, who is now consequently known as "the father of trigonometry."[3]
Sumerian astronomers introduced angle measure, using a...
...Trigonometry (from Greek trigōnon "triangle" + metron "measure"[1]) is a branch of mathematics that studies triangles and the relationships between the lengths of their sides and the angles between those 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.[2] It is also the foundation of the practical art of surveying.
Trigonometry basics are often taught in school either as a separate course or as part of a precalculus 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. Spherical 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.
Contents
f one angle of a triangle is 90 degrees and one of the other angles is known, the third is thereby fixed, because the three angles of any triangle add up to 180 degrees. The two acute angles therefore add up to 90 degrees: they are complementary angles. The shape of a triangle is completely...
...the definition of the trigonometric functions
sin β = b/c,
cos β = a/c.
As is well known, the sum of sin² β and cos² β is equal to 1:
sin² β + cos² β = 1,
which is a fundamental trigonometric identity. Consequently,
(b/c)² + (a/c)² = 1
implying a² + b² = c².
Critique
The identity sin² β + cos² β = 1 is indeed fundamental in trigonometry. However, its derivation is based on the Pythagorean theorem, to start with. Thus, the trigonometric "proof" above may well serve as an example of the circular reasoning, a vicious circle as the latter is sometimes referred to.
A Later Remark
As it comes out sin² β + cos² β = 1 admits a proof independent of the Pythagorean theorem but based solely on the subtraction formulas for sine and cosine. Thus my critique of the above proof should be taken with the grain of salt. It nonetheless can be used in a polemique. Any one who promotes the above proof with no knowledge as to how to derive sin² β + cos² β = 1 with no recourse to the Pythagorean theorem makes a severe logical mistake.
Proof 5
A mistake of a higher order is sometimes committed by more advanced students of mathematics who went beyond trigonometry and ventured into the multidimensional geometry. In multidimensional spaces whose elements are vectors, one often defines what is known as the scalar product and then also an angle between two vectors. Say, for two vectors a and b, if the scalar product is denoteda·b, then the angle γ between the...