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 determined, except for similarity, by the angles. Once the angles are known, the ratios of the sides are determined, regardless of the overall size of the triangle. If the length of one of the sides is known, the other two are determined. These ratios are given by the following trigonometric functions of the known angle A, where a, b and c refer to the lengths of the sides in the accompanying figure:
Sine function (sin), defined as the ratio of the side opposite the angle to the hypotenuse.
...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 division of...
...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...
...Amongst the lay public of nonmathematicians and nonscientists, trigonometry is known chiefly for its application to measurement problems, yet is also often used in ways that are far more subtle, such as its place in the theory of music; still other uses are more technical, such as in number theory. The mathematical topics of Fourier series and Fourier transforms rely heavily on knowledge of trigonometric functions and find application in a number of areas, including statistics.
There is an enormous number of applications of trigonometry and trigonometric functions. For instance, the technique of triangulation is used in astronomy to measure the distance to nearby stars, in geography to measure distances between landmarks, and in satellite navigation systems. The sine and cosine functions are fundamental to the theory of periodic functions such as those that describe sound and light waves.
Fields which make use of trigonometry or trigonometric functions include astronomy (especially, for locating the apparent positions of celestial objects, in which spherical trigonometry is essential) and hence navigation (on the oceans, in aircraft, and in space), music theory, acoustics, optics, analysis of financial markets, electronics, probability theory, statistics, biology, medical imaging (CAT scans and ultrasound), pharmacy, chemistry, number theory (and hence cryptology), seismology, meteorology, oceanography, many...
...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...
...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...
...SPHERICAL TRIGONOMETRY
DEFINITION OF TERMS
The sphere is the set of all points in a threedimensional space such that the distance of each from a fixed point is constant. The fixed point and the given distance are called the center and the radius of the sphere respectively.
The intersection of a plane with a sphere is a circle. If the plane passes through the center of the sphere, the intersection is a great circle; otherwise, the intersection is a small circle.
A line perpendicular to the plane of a circle and through the center of the sphere is called the axis of the circle. The intersection of this axis and the sphere are called the poles of the circle. Opposite ends of a diameter are identified as antipodal points. Two great circles intersecting in a pair of antipodal points divide the sphere into four regions called lunes. Thus a lune is bounded by the arcs of two great circles.
The polar distance (in angular units) of a circle is the least distance of a point on the circle to its pole.
Two distinct points on the sphere which are not ends of a diameter divide the great circle into two arcs. The shorter arc is called the minor arc.
SPHERICAL TRIANGLES
A spherical triangle is that part of the surface of a sphere bounded by three arcs of great circles. The bounding arcs are called the sides of the spherical triangle and the intersections of these arcs are called the vertices of the spherical triangle. The angle formed by two...
...Right Triangle TrigonometryTrigonometry is a branch of mathematics involving the study of triangles, and has applications in fields such as engineering, surveying, navigation, optics, and electronics. Also the ability to use and manipulate trigonometric functions is necessary in other branches of mathematics, including calculus, vectors and complex numbers. Rightangled Triangles In a rightangled triangle the three sides are given special names. The side opposite the right angle is called the hypotenuse (h) – this is always the longest side of the triangle. The other two sides are named in relation to another known angle (or an unknown angle under consideration).
If this angle is known or under consideration
h
θ
this side is called the opposite side because it is opposite the angle
This side is called the adjacent side because it is adjacent to or near the angle Trigonometric Ratios In a rightangled triangle the following ratios are defined sin θ = opposite side length o = hypotenuse length h cosineθ = adjacent side length a = hypotenuse length h
tangentθ =
opposite side length o = adjacent side length a
where θ is the angle as shown
These ratios are abbreviated to sinθ, cosθ, and tanθ respectively. A useful memory aid is Soh Cah Toa pronounced ‘socartowa’
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Unknown sides and angles in right angled triangles can be found using these ratios. Examples Find the value of the indicated unknown (side...