brief essay about the history of trigonometry

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• Introduction

Ancient Egypt and the Mediterranean world

• India and the Islamic world
• Passage to Europe
• From geometric to analytic trigonometry
• Application to science
• Trigonometric functions of an angle
• Tables of natural functions
• Plane trigonometry
• Spherical trigonometry
• Analytic trigonometry
• Polar coordinates
• Transformation of coordinates

trigonometry

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• Table Of Contents

trigonometry , the branch of mathematics concerned with specific functions of angles and their application to calculations. There are six functions of an angle commonly used in trigonometry. Their names and abbreviations are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). These six trigonometric functions in relation to a right triangle are displayed in the figure. For example, the triangle contains an angle A , and the ratio of the side opposite to A and the side opposite to the right angle (the hypotenuse) is called the sine of A , or sin A ; the other trigonometry functions are defined similarly. These functions are properties of the angle A independent of the size of the triangle, and calculated values were tabulated for many angles before computers made trigonometry tables obsolete. Trigonometric functions are used in obtaining unknown angles and distances from known or measured angles in geometric figures.

Trigonometry developed from a need to compute angles and distances in such fields as astronomy , mapmaking , surveying , and artillery range finding. Problems involving angles and distances in one plane are covered in plane trigonometry . Applications to similar problems in more than one plane of three-dimensional space are considered in spherical trigonometry .

History of trigonometry

Classical trigonometry.

The word trigonometry comes from the Greek words trigonon (“triangle”) and metron (“to measure”). Until about the 16th century, trigonometry was chiefly concerned with computing the numerical values of the missing parts of a triangle (or any shape that can be dissected into triangles) when the values of other parts were given. For example, if the lengths of two sides of a triangle and the measure of the enclosed angle are known, the third side and the two remaining angles can be calculated. Such calculations distinguish trigonometry from geometry , which mainly investigates qualitative relations. Of course, this distinction is not always absolute: the Pythagorean theorem , for example, is a statement about the lengths of the three sides in a right triangle and is thus quantitative in nature. Still, in its original form, trigonometry was by and large an offspring of geometry; it was not until the 16th century that the two became separate branches of mathematics .

Several ancient civilizations—in particular, the Egyptian , Babylonian , Hindu, and Chinese—possessed a considerable knowledge of practical geometry, including some concepts that were a prelude to trigonometry. The Rhind papyrus , an Egyptian collection of 84 problems in arithmetic , algebra , and geometry dating from about 1800 bce , contains five problems dealing with the seked . A close analysis of the text, with its accompanying figures, reveals that this word means the slope of an incline—essential knowledge for huge construction projects such as the pyramids . For example, problem 56 asks: “If a pyramid is 250 cubits high and the side of its base is 360 cubits long, what is its seked ?” The solution is given as 5 1 / 25 palms per cubit , and, since one cubit equals 7 palms, this fraction is equivalent to the pure ratio 18 / 25 . This is actually the “run-to-rise” ratio of the pyramid in question—in effect, the cotangent of the angle between the base and face. It shows that the Egyptians had at least some knowledge of the numerical relations in a triangle, a kind of “proto-trigonometry.”

Trigonometry in the modern sense began with the Greeks . Hipparchus ( c. 190–120 bce ) was the first to construct a table of values for a trigonometric function . He considered every triangle—planar or spherical—as being inscribed in a circle, so that each side becomes a chord (that is, a straight line that connects two points on a curve or surface , as shown by the inscribed triangle A B C in the figure). To compute the various parts of the triangle, one has to find the length of each chord as a function of the central angle that subtends it—or, equivalently, the length of a chord as a function of the corresponding arc width. This became the chief task of trigonometry for the next several centuries. As an astronomer, Hipparchus was mainly interested in spherical triangles, such as the imaginary triangle formed by three stars on the celestial sphere , but he was also familiar with the basic formulas of plane trigonometry. In Hipparchus’s time these formulas were expressed in purely geometric terms as relations between the various chords and the angles (or arcs) that subtend them; the modern symbols for the trigonometric functions were not introduced until the 17th century.

Trigonometry/A Brief History of Trigonometry

The Babylonians could measure angles, and are believed to have invented the division of the circle into 360º. [1] However, it was the Greeks who are seen as the original pioneers of trigonometry.

A Greek mathematician, Euclid , who lived around 300 BC was an important figure in geometry and trigonometry. He is most renowned for Euclid's Elements , a very careful study in proving more complex geometric properties from simpler principles. Although there is some doubt about the originality of the concepts contained within Elements , they're influential in how we think about proofs and geometry today; indeed, it has been said that the Elements have "exercised an influence upon the human mind greater than that of any other work except the Bible. [2]

• 1 First Tables of Sines or Cosines
• 2 The Pythagorean Theorem
• 3 Heron's Formula
• 4 References

First Tables of Sines or Cosines

In the second century BC a Greek mathematician, Hipparchus , is thought to have been the first person to produce a table for solving a triangle's lengths and angles. [3]

The Pythagorean Theorem

The Pythagorean theorem is named after the Greek mathematician Pythagoras, who by tradition is credited with its discovery and proof, [4] [5] although it is often argued that knowledge of the theorem predates him. There is much evidence that Babylonian mathematicians understood the formula. [6]

Heron's Formula

And, for a cyclic quadrilateral (one whose all 4 sides lie inside a circle), this formula can be used:-

The formula is believed to be due to Heron of Alexandria (10 – 70 AD), a Greek mathematician. The formula has nothing to do with the Heron (a type of bird).

• ↑ http://www.lscc.edu/faculty/steven_a_boast/Shared%20Documents/MAC%201114%20Trigonometry/Activities%20and%20Labs/MAC%201114%20Measuring%20Angles%20Lab.pdf
• ↑ "Hipparchus of Rhodes" . 1999.
• ↑ George Johnston Allman (1889). Greek Geometry from Thales to Euclid (Reprinted by Kessinger Publishing LLC 2005 ed.). Hodges, Figgis, & Co. p. 26. ISBN   143260662X . The discovery of the law of three squares, commonly called the "theorem of Pythagoras" is attributed to him by – amongst others – Vitruvius, Diogenes Laertius, Proclus, and Plutarch ...
• ↑ ( Heath 1921 , Vol I, p. 144) harv error: no target: CITEREFHeath1921 ( help )
• ↑ Otto Neugebauer (1969). The exact sciences in antiquity (Republication of 1957 Brown University Press 2nd ed.). Courier Dover Publications. p. 36. ISBN   0486223329 . . For a different view, see Dick Teresi (2003). Lost Discoveries: The Ancient Roots of Modern Science . Simon and Schuster. p. 52. ISBN   074324379X . , where the speculation is made that the first column of a tablet 322 in the Plimpton collection supports a Babylonian knowledge of some elements of trigonometry. That notion is pretty much laid to rest by Eleanor Robson (2002). "Words and Pictures: New Light on Plimpton 322" . The American Mathematical Monthly . Mathematical Association of America. 109 (2): 105–120. doi : 10.2307/2695324 . {{ cite journal }} : Invalid |ref=harv ( help ) See also pdf file . The accepted view today is that the Babylonians had no awareness of trigonometric functions. See Abdulrahman A. Abdulaziz (2010). "The Plimpton 322 Tablet and the Babylonian Method of Generating Pythagorean Triples" . ArXiv preprint . {{ cite journal }} : Invalid |ref=harv ( help ) §2, page 7.

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Historical Development of Trigonometry

• 12 January 2024

Introduction to Development of Trigonometry

Understanding the development of trigonometry is paramount for anyone delving into the realms of mathematics, science, and engineering. Trigonometry has undergone a fascinating journey of development over the centuries. By tracing its historical roots and the contributions of luminaries, one gains profound insights into the cultural, intellectual, and scientific advancements of diverse societies.

Moreover, comprehending the historical development of trigonometry allows individuals to appreciate the interconnectedness of mathematical concepts and witness how this branch of mathematics evolved to address practical problems in navigation, astronomy, and architecture. Studying its historical context not only fosters a deeper understanding of the subject but also highlights the universality and timeless relevance of mathematical principles, fostering an appreciation for the rich tapestry of human intellectual achievement over time.

Greek Contributions to Trigonometry

Modern trigonometry traces its earliest roots to ancient Greek mathematicians who began studying the relationships between chords, arcs and angles in circles.

The concept of “chords” emerged as straight line segments within circular geometries that subtended an angle and intersected the endpoints of an arc on the circumference. Greek geometers, including notable names like Aristarchus, produced some of the first tables documenting chords of various arc lengths.

The ancient Greeks, including mathematicians like Hipparchus and Ptolemy, further expanded upon the ideas of Pythagoras early work and developed early trigonometric concepts. The work of these later mathematicians, along with the foundational principles established by Pythagoras, contributed to the evolution of trigonometry in subsequent centuries.

To learn about life of Pythagoras and his contributions in mathematics, read The Life And Philosophy of Pythagoras .

This early chord analysis was significantly advanced around 150 BCE by the influential Greek astronomer-mathematician Hipparchus of Nicaea, who greatly expanded the scope of existing chord tables.

Nearly three centuries later, the Greco-Roman scholar Claudius Ptolemy expanded further on Hipparchus’s foundations in his enormously impactful astronomical text the Almagest, which listed a thorough table of chords systematically organized by central angles. Ptolemy’s meticulously calculated tables transmitted classical knowledge that informed centuries of future mathematicians. To learn more about the relation of astronomy and mathematics, read Role of Astronomy in Mathematics

Trigonometry and Muslim Astronomy

As civilizations built upon classical Greek mathematical astronomy, trigonometric techniques became further intertwined with predicting celestial cycles and movements. Muslim scholars began constructing extensive trigonometric tables detailing chords, arcs and related quantities to facilitate precise calculation for calendars, navigation and mapping the heavens.

For example, around 1000 CE the brilliant Persian polymath al-Biruni advanced trigonometry significantly through building sine and tangent tables more accurate than his predecessors while writing extensively on theoretical underpinnings.

Influenced by al-Biruni’s work, the 13th century astronomer and mathematician Nasir al-Din al-Tusi made his most notable contribution by compiling the most extensive trigonometric table of his era as part of his famous Ilkhani Tables, which tabulated trigonometric values correct to 8 decimal places through using techniques like Ptolemy’s Theorem.

To learn about decimals, read How To Do Basic Operations With Decimals .

This medieval period witnessed profound refinement of trigonometry deeply linked with the pressing needs of astronomers, cementing foundations for early modern breakthroughs.

Modern Trigonometry Revolution

As mathematics transitioned towards symbolic algebra from its geometric heritage, concerted efforts to define trigonometry functionally also catalyzed modernization.

In the 16th century, French mathematician François Viète introduced an influential approach using variables to represent unknowns while computing values relying on a mix of traditional rhetorical and modern symbolic styles. Viète helped set the stage for conceptually rethinking trigonometry in algebraic terms rather than solely geometric constructs.

To learn about basic algebra, read Introductory Algebra .

Building on this, the great 18th century Swiss mathematician Leonhard Euler formally defined the six primary trigonometric functions still used today – sine, cosine, tangent, cotangent, secant and cosecant. This established a normative technical vocabulary and notational system adopted internationally. Euler also made key advances in establishing trigonometric identities that unlocked new discovery through equation equivalencies.

By regimenting terminology, clarifying functional relationships, and revealing deeper truths via identities, pioneering early modern scholars transformed trigonometry towards today’s function-based field essential across sciences and engineering.

Trigonometry in Calculus – Interlink that Lead to Growth

While trigonometry already reached advanced maturity by the 1700s, seminal insights revealing hidden ties binding trigonometric functions and exponential relations through calculus vastly expanded the field’s utility and influence.

Mathematicians like Leonhard Euler and Joseph Lagrange established an intimate calculus connection: the derivative of sine and cosine expressions produced new functions definable only through differential techniques. This derivative link showcased foundational roles for exponentials underlying trigonometry’s structure.

Calculus also provided tools to generalize modeling any cyclical phenomena beyond simple harmonic motion by applying trigonometric building blocks in novel combinations and extensions. Powerful integration methods enabled tackling trigonometric equations insoluble for classical geometers.

Interweaving trigonometry and groundbreaking infinitesimal analysis through calculus precipitated maturity for both disciplines. For example, Euler’s prolific Introduction to Analysis of the Infinite laid rigorously formalized foundations for calculus leveraging trigonometric representations of functions.

Unforeseen vistas emergent from calculus synthesis propelled trigonometry’s progress towards indispensable modern roles across science and engineering.

Modern Trigonometry in the 20th Century

Major twentieth century advances propelled the Development of Trigonometry and diversification as a central mathematical discipline.

New computational and visual mapping tools like plotting trigonometric functions on coordinate planes over continuous domains rather than at discrete tabulated values enabled unprecedented dynamic modeling and analysis. Software automation increased feasibility of calculating expansive high-precision tables enumerating trigonometric values at scale.

Theoretically, groundbreaking connections with complex numbers and non-Euclidean geometries spawned hybrid subfields like complex trigonometry with applications from physics to electrical engineering. Later pioneers like Russian mathematician D. K. Faddeev even formalized set valued functions expanding into multidimensional trigonometry.

So alongside preserving classical roles in science and surveying, innovative extensions into more versatile function representations, applied visualizations, and special sub-domains underscored the vitality of contemporary trigonometry research even through recent times.

If you’re interested in learning about Number theory, try reading Number Theory in Cryptography

Conclusion – Development of Trigonometry 

Solution of ancient astronomical problems is the fruit of Development of Trigonometry. Trigonometry’s universal relevance magnifying across recent centuries cements standing among most fundamentally applicable mathematical disciplines. 

Trigonometric functions and techniques enable analysis essential for architecture, geography and navigation by simplifying complex calculations. Electrical engineers rely extensively on trigonometric identities for designing circuits and filtering signals. Economists apply trigonometric time series forecasting and detecting cyclical financial patterns.

Additionally, trigonometry retains intrinsic value powering traditional pursuits like optics while underpinning vanguard directions like aerospace orbital predictions – now feasible through computational access to high precision tables.

By abstracting nature’s pervasive episodic cycles into lucid analytical tools, trigonometry empowers interdisciplinary innovation and reveals enduring harmonic order suffusing the cosmos. As a basis for fields from calculus to complex dynamics, trigonometry remains a keystone of mathematical science.

Frequently Asked Questions (FAQs)

Why did islamic astronomers invest in the development of trigonometry.

Islamic astronomers invested in the development of trigonometry for several reasons such as trigonometric tables facilitated precise calculations for calendars, helping determine important religious events and marking the passage of time accurately.

Who is the father of trigonometry?

Hipparchus is commonly referred to as the father of trigonometry, it’s important to note that the development of trigonometric concepts involved contributions from multiple ancient cultures, including the Babylonians and Egyptians. Greek mathematicians, including Hipparchus, further refined and expanded upon these ideas, shaping the early understanding of trigonometry.

Where did the history of trigonometry originated?

The history of trigonometry is thus a composite of contributions from various cultures and periods, each building upon the knowledge of its predecessors. The field evolved over centuries, with different civilizations adding their insights and applications such as Ancient Greeks, Egyptians, Muslim Mathematicians and Europe and the Middle Ages.

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A brief history of trigonometry.

• 1800 BCE -------------- Babylon -- Plimpton 322 Tablet contains a list of Pythagorean triplets more than a thousand years before Pythagoras and his formula
• 1550 BCE -------------- Egypt ----- Rhind Mathematical Papyrus features example problems calculating the slope, or Seked, of a pyramid
• 500s BCE ---------------Greece --- Pythagoras is said to have invented the famous theorem
• 200 BCE to 200 CE -- Greece ---- Hipparchus, Menelaus, and Ptolemy each make additions and revisions to tables of chords for use in astronomy
• 500 CE ----------------- India ------ First Sine Tables appear, begin to overtake chord tables
• 1000 CE --------------- Islam ------ Mathematicians are using all six Trigonometric functions
• 1765 CE --------------- England -- Turner publishes book on Trigonometry

The Plimpton 322 Clay Tablet, curently housed in the Plimpton collection at Columbia University.  

The rhind mathematical papyrus, currently located in the british museum..

     Trigonometry traces its roots back to Ancient Babylon and Egypt.  This is because the oldest surviving pieces of evidence for trigonometry are the Plimpton Tablet #322 and the Rhind Mathematical Papyrus.  The Plimpton Tablet dates from around 1800 BCE and contains a table of Pythagorean triplets, or numbers that satisfy the sides of a right triangle [1].  The Rhind Papyrus, dating from around 1550 BCE, features problems solving the slope of a pyramid, called a Seked [2].

     The Ancient Greeks heavily contributed to trigonometry, with Hipparchus, Menelaus, and Ptolemy each living after one another and building off the chord tables of their predecessors [3].  Originally, the side of a triangle in question would be called a chord because it intercepted the arc of its corresponding angle.  The Greeks were mostly concerned with the sky and the heavens.  Therefore, Trigonometry started by studying the positions of the stars.  Hipparchus is said to be the founder of Trigonometry, and Ptolemy wrote the Almagest, an important work on the subject [4].  However, the first use of Sine tables was recorded instead in India in the 6th century and spread back west [5].  By the height of the Medieval Islamic World, mathematicians there were using the six trigonometric functions we know today, namely, the Sine, Cosine, Tangent, Cosecant, Secant, and Cotangent [6].

Works Cited:

Daniel f. mansfield and n.j. wildberger, "plimpton 322 is babylonian exact sexagesimal trigonometry,"  historia mathematica  44, no. 4 (aug. 2017): 395, https://doi.org/10.1016/j.hm.2017.08.001., glen van brummelen,  the mathematics of the heavens and the earth: the early history of trigonometry , (princeton, nj: princeton university press, 2009), 11., morris kline,  mathematical thought from ancient to modern times , (new york, ny: oxford university press, 1972), 119-122., van brummelen, 95., david m. bressoud, “historical reflections on teaching  trigonometry,”  the mathematics teacher  104, no. 2 (sept. 2010): 110., image sources:, unknown, plimpton 322 , 1800 bce, sept. 13, 2006, wikimedia commons , accessed dec. 12, 2017, http://www.math.ubc.ca/~cass/courses/m446-03/pl322/pl322.html., paul james cowie, rhind mathematical papyrus , 1550 bce, june 4, 2017, wikimedia commons , accessed dec. 12, 2017, https://commons.wikimedia.org/wiki/file:rhind_mathematical_papyrus.jpg..

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Trigonometry: A Very Short Introduction

Coordinator of Mathematics

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Trigonometry: A Very Short Introduction draws together the full history of trigonometry, stretching across two millennia and several cultures such as ancient Greece, medieval India, and the Islamic world. It introduces the key concepts of trigonometry, drawing readers beyond the basic relationships first encountered in school to reveal the richness of the entire subject of trigonometry and ideas such as curved space. It also explores connections with genuine modern applications, including navigation, the analysis of music, computer graphics, and powerful modelling tools in science, and shows how trigonometry has participated in big questions about the world, including the shape of the universe and the nature of infinity.

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• Social Sciences

What Is Trigonometry? Definition, History, and Uses

• Author: Melanie Palen

Trigonometry is more than just measuring triangles. You'll encounter circles, hyperbolae, and ellipses, too!

Melanie Shebel

What Is Trig?

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles, and the calculations based on them. It is used in a wide range of fields, including astronomy, engineering, physics, and navigation.

The basic concepts of trigonometry include the six trigonometric functions: sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc). These functions are defined in terms of the ratios of the sides of a right triangle (a triangle with one 90-degree angle) and are usually denoted by the corresponding lowercase letters.

Trigonometry can be used to solve problems involving angles and distances, such as finding the height of a building or the distance to a star, and can also be used to model and analyze periodic phenomena, such as waves and oscillations.

Trigonometry, a brief description. Triangles and circles and hyberbolae, oh my!

The Definition of Trigonometry Trigonometry is the branch of mathematics devoted explicitly to the relationship between the sides and angles of triangles. Its name even sounds triangle-y and for a good reason: the word trigonometry is derived from two Greek words - 'trigonon' which means 'triangle' and 'metron' meaning 'measure.' Put together, the words mean "triangle measuring."

Trig Is More Than Triangles!

Trigonometry is more than just measuring triangles. It's also circle measuring, hyperbola measuring, and ellipse measuring—things that are decidedly very non-triangular. This can be achieved by the usage of the ratios between the sides and angles of a triangle (which will be discussed later) and the manipulation of variables.

A part of the Rhind Mathematical Papyrus showing early trigonometry

public domain

The Early Roots of Trigonometry

Defining the very beginning of a concept is difficult. Because mathematics is so abstract, we can't just say a cave painting of a triangle is trigonometry. What did the painter mean by the triangle? Did he just like triangles? Was he enthralled with how the length of one side, another side, and the angle they made dictated the length and angles of the other sides?

Furthermore, paperwork back in the day was notoriously poorly filed and sometimes burned. Also, duplicates were often not made (they didn't have electricity to power copy machines.) In short, stuff got lost.

The earliest known "strong" example of trigonometry is found on the Rhind Mathematical Papyrus which dates to around 1650 BC. The second book of the papyrus shows how to find the volume of cylindrical and rectangular granaries and how to find the area of a circle (which at that time was approximating using an octagon.) Also on the papyrus are calculations for pyramids including a sophisticated approach that uses a beat-around-the-bush method for finding the value of the cotangent of the angle to a pyramid's base and its face.

In the late 6th century BC, the Greek mathematician Pythagoras gave us:

a 2 + b 2 = c 2

The stands as one of the most commonly used relations in trigonometry and is a special case for the Law of Cosines:

c 2 = a 2 + b 2 - 2ab cos(θ)

However, the systematic study of trigonometry dates to the middle ages in Hellenistic India where it began to spread across the Greek empire and bled into Latin territories during the Renaissance. With the Renaissance came an enormous growth of mathematics.

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However, it wasn't until the 17th and 18th centuries that we saw the development of modern trigonometry with the likes of Sir Isaac Newton and Leonhard Euler (one of the most significant mathematicians the world will ever know). It is Euler's formula that establishes the fundamental relationships between the trigonometric functions.

The trig functions graphed

The Trigonometric Functions

In a right triangle, six functions can be used to relate the lengths of its sides with an angle (θ.)

The three ratios sine, cosine, and tangent are reciprocals of the ratios cosecant, secant, and cotangent respectively, as shown:

The three ratios sine, cosine, and tangent are reciprocals of the ratios cosecant, secant, and cotangent respectively, as shown.

If given the length of any two sides, the usage the Pythagorean Theorem not only allows one to find the length of the missing side of the triangle but the values for all six trigonometric functions.

While the usage of the trigonometric functions may seem limited (one might only need to find the unknown length of a triangle in a small number of applications), these tiny pieces of information can be extended much further. For example, right triangle trigonometry can be used in navigation and physics.

For example, sine and cosine can be used to resolve polar coordinates to the Cartesian plane, where x = r cos θ and y = r sin θ .

Using a right triangle to define a circle.

Pbroks13, cc-by-sa, via Wikimedia Commons

Geometric Curves: Conics in Trig

As mentioned above, trigonometry is powerful enough to make measurements of things that are not triangles. Conics such as hyperbolae and ellipses are examples of how awesomely sneaky trigonometry can be—a triangle (and all its formulae) can be hidden inside an oval!

Let's start with a circle. One of the first things one learns in trigonometry is that the radii and arcs of a circle can be found using a right triangle. This is because the hypotenuse of a right triangle is also the slope of the line connecting the center of the circle with a point on the circle (as shown below). This same point can also be found using the trigonometric functions.

Working with triangles to find information about a circle is easy enough, but what happens with ellipses? They’re just flattened circles, but the distance from the center to the edge is not uniform as it is in a circle.

It could be argued that an ellipse is better defined by its foci than its center (while noting that the center is still useful in calculating the equation for the ellipse). The distance from one focus (F1) to any point (P) added to the distance from the other focus (F2) to point P does not differ as one travels around the ellipse. An ellipse is related using b2 = a2 – c2 where c is the distance from the center to either focus (either positive or negative), a is the distance from the center to the vertex (major axis), and b is the distance from the center to the minor-axis.

Equations for Ellipses

The equation for an ellipse with center (h,k) where the x-axis is the major axis (as in the ellipse shown below) is:

An ellipse where the x-axis is the major axis. Vertices at (h,a) and (h,-a).

However, the equation for an ellipse where the major axis is the y-axis is related by:

Equations for Hyperbolae

A hyperbola looks very different from an ellipse. In fact, nearly oppositely so . . . it’s a hyperbola split in half with the halves facing in opposite directions. However, in terms of finding the equations of hyberbolae versus any other “shape,” the two are closely related.

A hyperbola transversed across the x-axis

For x-axis transversed hyperbolae

For y-axis transversed hyperbolae

Like an ellipse, the center of a hyperbola is referenced by (h,k.) However, a hyperbola only has one vertex (noted by the distance a from the center in either the x or y-direction depending on the transverse axis.)

Also unlike an ellipse, the foci of a hyperbola (noted by distance c from the center) are further from the center than the vertex. The Pythagorean Theorem rears its head here too, where c2 = b2 + a2 using the equations to the right.

As you can see, trigonometry can bring one further than just finding the missing length of a triangle (or a missing angle.) It's used for more than just measuring the height of a tree by the shadow it casts or finding the distance between two buildings given some unusual scenario. Trigonometry can be applied further to define and describe circles and circle-like shapes.

Hyperbolae and ellipses serve as great examples of how trigonometry can quickly deviate from just stating the Pythagorean Theorem and the few relationships between the lengths of the sides of a simple triangle (the trig functions.)

The toolset of equations in trigonometry is small, however, with a bit of creativity and manipulation, these equations can be used to obtain an accurate description of a wide variety of shapes such as ellipses and hyperbolae.

This article is accurate and true to the best of the author’s knowledge. Content is for informational or entertainment purposes only and does not substitute for personal counsel or professional advice in business, financial, legal, or technical matters.

© 2017 Melanie Palen

Peggy Woods from Houston, Texas on July 07, 2017:

I never studied trigonometry and did not need it for a nursing degree. My husband took it however. I had no idea that the concept was used so many centuries ago! Very interesting!

Rajan Singh Jolly from From Mumbai, presently in Jalandhar, INDIA. on July 02, 2017:

Nice introduction to trignometry. I wonder why this seemed too difficult when I had my first brush with it in my first year of college?

Polynomial Rules: What Defines Polynomials?

Reciprocal Identities in Trigonometry (With Examples)

Trigonometry: Graphing the Sine, Cosine and Tangent Functions

Cofunction Identities in Trigonometry (With Proof and Examples)

Multiply Polynomials (With Examples): FOIL & Grid Methods

Pythagoras vs. Trigonometry: Which One Should I Use?

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The Historical Roots of Trigonometry

Mathematical Pioneers in Trigonometry

Trigonometry in the modern world, conclusion: a timeless legacy.

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Trigonometry

Trigonometry (from Greek Τριγωνομετρία "tri = three" + "gon = angle" + "metr[y] = to measure") is a branch of mathematics that deals with triangles, particularly those plane triangles in which one angle has 90 degrees (right triangles) . Trigonometry deals with relationships between the sides and the angles of triangles and with the trigonometric functions, which describe those relationships.

Trigonometry has applications in both pure mathematics and in applied mathematics, where it is essential in many branches of science and technology. It is usually taught in secondary schools either as a separate course or as part of a precalculus course. Trigonometry is informally called "trig" or "trigo."

• 2.1 Extending the definitions
• 2.2 Mnemonics
• 2.3 Calculating trigonometric functions
• 3 Applications of trigonometry
• 4.1.1 Pythagorean identities
• 4.1.2.1 Sum to product
• 4.1.2.2 Product to sum [4]
• 4.1.2.3 Sine, cosine, and tangent of a sum
• 4.1.3 Half-angle identities
• 4.1.4 Stereographic (or parametric) identities
• 4.2.1 Law of sines
• 4.2.2 Law of cosines
• 4.2.3 Law of tangents
• 6 References
• 7 External links

A branch of trigonometry, called spherical trigonometry, studies triangles on spheres , and is important in astronomy and navigation .

Trigonometry was developed for use in sailing as a navigation method used with astronomy . [1] The origins of trigonometry can be traced to the civilizations of ancient Egypt , Mesopotamia and the Indus Valley ( India ), more than 4000 years ago. The common practice of measuring angles in degrees, minutes and seconds comes from the Babylonian 's base sixty system of numeration.

The first recorded use of trigonometry came from the Hellenistic mathematician Hipparchus [2] c. 150 B.C.E. , who compiled a trigonometric table using the sine for solving triangles. Ptolemy further developed trigonometric calculations c. 100 C.E.

The ancient Sinhalese in Sri Lanka , when constructing reservoirs in the Anuradhapura kingdom, used trigonometry to calculate the gradient of the water flow. Archeological research also provides evidence of trigonometry used in other unique hydrological structures dating back to 4 B.C.E.

The Indian mathematician Aryabhata in 499, gave tables of half chords which are now known as sine tables, along with cosine tables. He used zya for sine, kotizya for cosine, and otkram zya for inverse sine, and also introduced the versine. Another Indian mathematician, Brahmagupta in 628, used an interpolation formula to compute values of sines, up to the second order of the Newton -Stirling interpolation formula.

In the tenth century, the Persian mathematician and astronomer Abul Wáfa introduced the tangent function and improved methods of calculating trigonometry tables. He established the angle addition identities, for example, sin ( a + b ), and discovered the sine formula for spherical geometry:

Also in the late tenth and early eleventh centuries, the Egyptian astronomer Ibn Yunus performed many careful trigonometric calculations and demonstrated the formula

Detailed methods for constructing a table of sines for any angle were given by the Indian mathematician Bhaskara in 1150, along with some sine and cosine formulae. Bhaskara also developed spherical trigonometry.

The thirteenth century Persian mathematician Nasir al-Din Tusi, along with Bhaskara, was probably the first to treat trigonometry as a distinct mathematical discipline. Nasir al-Din Tusi in his Treatise on the Quadrilateral was the first to list the six distinct cases of a right angled triangle in spherical trigonometry.

In the fourteenth century, Persian mathematician al-Kashi and Timurid mathematician Ulugh Beg (grandson of Timur ) produced tables of trigonometric functions as part of their studies of astronomy.

The mathematician Bartholemaeus Pitiscus published an influential work on trigonometry in 1595 which may have coined the word "trigonometry" itself.

If 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 right triangle is completely determined, up to similarity, by the angles. This means that once one of the other angles is known, the ratios of the various sides are always the same regardless of the overall size of the triangle. 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:

• The sine function (sin), defined as the ratio of the side opposite the angle to the hypotenuse.

• The cosine function (cos), defined as the ratio of the adjacent leg to the hypotenuse.

• The tangent function (tan), defined as the ratio of the opposite leg to the adjacent leg.

The hypotenuse is the side opposite to the 90 degree angle in a right triangle; it is the longest side of the triangle, and one of the two sides adjacent to angle A . The adjacent leg is the other side that is adjacent to angle A . The opposite side is the side that is opposite to angle A . The terms perpendicular and base are sometimes used for the opposite and adjacent sides respectively. Many people find it easy to remember what sides of the right triangle are equal to sine, cosine, or tangent, by memorizing the word SOH-CAH-TOA (see below under Mnemonics ).

The reciprocals of these functions are named the cosecant (csc or cosec), secant (sec) and cotangent (cot), respectively. The inverse functions are called the arcsine, arccosine, and arctangent, respectively. There are arithmetic relations between these functions, which are known as trigonometric identities.

With these functions one can answer virtually all questions about arbitrary triangles by using the law of sines and the law of cosines. These laws can be used to compute the remaining angles and sides of any triangle as soon as two sides and an angle or two angles and a side or three sides are known. These laws are useful in all branches of geometry, since every polygon may be described as a finite combination of triangles.

Extending the definitions

The above definitions apply to angles between 0 and 90 degrees (0 and π/2 radians) only. Using the unit circle, one can extend them to all positive and negative arguments (see trigonometric function). The trigonometric functions are periodic, with a period of 360 degrees or 2π radians, except for the tangent and the cotangent, which have π as smallest period. This means their values repeat at those intervals.

The trigonometric functions can be defined in other ways besides the geometrical definitions above, using tools from calculus and infinite series. With these definitions the trigonometric functions can be defined for complex numbers. The complex function cis is particularly useful

See Euler's and De Moivre's formulas.

Students often use mnemonics to remember facts and relationships in trigonometry. For example, the sine , cosine , and tangent ratios in a right triangle can be remembered by representing them as strings of letters, as in SOH-CAH-TOA.

Alternatively, one can devise sentences which consist of words beginning with the letters to be remembered. For example, to recall that Tan = Opposite/Adjacent, the letters T-O-A must be remembered. Any memorable phrase constructed of words beginning with the letters T-O-A will serve.

It is of ethnographic interest to note that the mnemonic TOA-CAH-SOH can be translated in the local Singaporean Hokkien dialect to "big-legged woman," serving as an additional learning aid for students in Singapore . [3]

Another type of mnemonic describes facts in a simple, memorable way, such as "Plus to the right, minus to the left; positive height, negative depth," which refers to trigonometric functions generated by a revolving line.

Calculating trigonometric functions

Trigonometric functions were among the earliest uses for mathematical tables. Such tables were incorporated into mathematics textbooks and students were taught to look up values and how to interpolate between the values listed to get higher accuracy. Slide rules had special scales for trigonometric functions.

Today scientific calculators have buttons for calculating the main trigonometric functions (sin, cos, tan and sometimes cis) and their inverses. Most allow a choice of angle measurement methods, degrees, radians and, sometimes, Grad. Most computer programming languages provide function libraries that include the trigonometric functions. The floating point unit hardware incorporated into the microprocessor chips used in most personal computers have built in instructions for calculating trigonometric functions.

Applications of trigonometry

There are 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 physical sciences, land surveying and geodesy, architecture , phonetics, economics, electrical engineering , mechanical engineering , civil engineering , computer graphics , cartography , crystallography , and game development.

Common formulae

Certain equations involving trigonometric functions are true for all angles and are known as trigonometric identities. Many express important geometric relationships. For example, the Pythagorean identities are an expression of the Pythagorean Theorem. Here are some of the more commonly used identities, as well as the most important formulae connecting angles and sides of an arbitrary triangle. For more identities see trigonometric identity.

Trigonometric identities

Pythagorean identities.

Sum and product identities

Sum to product.

Product to sum [4]

Sine, cosine, and tangent of a sum

Half-angle identities

Stereographic (or parametric) identities

Triangle identities

In the following identities, A, B, and C are the angles of a triangle and a, b, and c are the lengths of sides of the triangle opposite the respective angles.

Law of sines

The law of sines (also know as the "sine rule") for an arbitrary triangle states:

where R is the radius of the circumcircle of the triangle.

Law of cosines

The law of cosines (also known as the cosine formula, or the "cos rule") is an extension of the Pythagorean theorem to arbitrary triangles:

or equivalently:

Law of tangents

The law of tangents:

• ↑ Christopher M. Linton, From Eudoxus to Einstein: A History of Mathematical Astronomy (New York, NY: Cambridge University Press, 2004, ISBN 978-0521045711 ).
• ↑ Joseph Hunt, The Beginnings of Trigonometry Rutgers . Retrieved December 16, 2021.
• ↑ Trigonometry—My Grandmother Also Can Do This! Diary of a Private O Level Maths Tutor in Singapore. Retrieved December 16, 2021.
• ↑ Eric W. Weisstein, Trigonometric Addition Formulas Wolfram MathWorld . Retrieved December 16, 2021.

References ISBN links support NWE through referral fees

• Hill, Tim. Essential Trigonometry: A Self-Teaching Guide . Questing Vole Press, 2013. ISBN 978-1937842161
• Lial, Margaret L., John Hornsby, and David I. Schneider. Trigonometry, 9th ed. Boston, MA: Pearson/Addison-Wesley, 2008. ISBN 0321528859 .
• Linton, Christopher M. From Eudoxus to Einstein: A History of Mathematical Astronomy. Cambridge, UK: Cambridge University Press, 2004. ISBN 978-0521045711 .
• Wesner, Terry H. Trigonometry, with applications . Wm. C. Brown, 1994. ISBN 978-0697122926

External links

All links retrieved May 2, 2023.

• Index for Trigonometry Mathwords.com .
• Trigonometry.
• Dave's Short Course in Trigonometry by David Joyce of Clark University.
• SOHCAHTOA Wolfram MathWorld .

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Published 2010

History of Trigonometry - Part 3

Pedagogical notes, teaching trigonometry: concepts and historical development., motivations and conventions, models, analogies and activities, fundamental concepts.

Trigonometry, derived from the Greek words "trigonon" ("triangle") and "metron" ("to measure"), focused on computing numerical values of triangle parts or shapes disassembled into triangles. Other parts' values were given until the 16th Century. Pythagoras, Aristotle, or the ancient civilization of the Greeks, Egyptians, and Babylonians may come to mind while looking at the dawn of trigonometry.

Pythagoras, the author of the Pythagorean Theorem, was afraid of beans and had a cult to worship triangles, his followers were not only vegan but also didn't eat beans because they believed that a part of their soul escaped every time, they passed gas.

The Pythagorean Theorem, traditionally known as the square of the hypotenuse's length equalling the sum of squares of the other two sides of a right-angled triangle, formulates as a² + b² = c². For example, (3)² + (4)² = (5)².

The Rhind papyrus (1800 BCE) was an Ancient Egyptian Collection of 84 mathematical problems to the likes as Arithmetic, Algebra, and Geometry. The Geometry portion also consisted of five issues dealing with the seked or what we now refer to as the hypotenuse.

The Rhind papyrus' 56th problem addressed a pyramid with a height of 250 cubits and a length of 360 cubits. The solution provided the ratio of 51/25 palms per cubit, equivalent to the pure ratio 18/25 considering that one cubit equals 7 palms.

This referred to the “run-to-rise” ratio of the pyramid in question—in effect, the cotangent of the angle between the base and face, from this we can examine the Egyptians' extent of knowledge of the numerical relations in a triangle along with trigonometry.

The Greeks initiated modern trigonometry, and Hipparchus (120 BCE) formulated the table of values for trigonometric functions. Being an astronomer, he focused on spherical triangles formed by three stars on the celestial sphere. Still, he was also familiar with the basic formulas of plane trigonometry.

The symbols for trigonometry were formulated only in the 17 th Century. Ptolemy’s Almagest was the first major ancient work on trigonometry to reach Europe that uses some elementary trigonometry that led to Ptolemy’s geocentric system being a succession of the heliocentric system of Nicolaus Copernicus.

The Babylonians used the Pythagorean Theorem some 1,000 years before Pythagoras was born. They wrote it down on a tablet now known as Plimpton 322.

The ancient Egyptians, Chinese, and Indians all used versions of its centuries before Pythagoras. An ancient Babylonian tablet used the Pythagorean Theorem 1200 years before Pythagoras was born.

India and the Islamic world were also great contributors to trigonometry which makes you wonder about the true importance and value of Mathematics as recent research shows us that 4000 years ago Neanderthals also had a basic sense of Mathematics.

FAQ: History of Trigonometry

Trigonometry has a rich history that dates back to ancient civilizations. It originated in ancient Greece and India, with early contributions from mathematicians like Hipparchus and Aryabhata. These ancient scholars developed trigonometric concepts to solve problems related to astronomy and geometry. Three important points about the origin of trigonometry: 1. Ancient Mesopotamia: The earliest known trigonometric concepts can be found in the ancient Mesopotamian civilization, around 2000 BCE. They used basic trigonometry to solve problems related to construction and land measurement. 2. Ancient Egypt: Ancient Egyptians also employed trigonometric principles, particularly in their architectural and engineering projects such as the construction of pyramids. They relied on geometric methods for measuring angles and distances. 3. Ancient Greece: The Greeks, notably mathematicians like Hipparchus and Ptolemy, made significant contributions to trigonometry. They developed more formalized methods and introduced the concept of chords, which laid the foundation for modern trigonometric functions.

Trigonometry underwent significant development in ancient India, where scholars made pioneering contributions to the field. Three important points about the evolution of trigonometry in ancient India: 1. Aryabhata's Contributions: Aryabhata, an Indian mathematician and astronomer from the 5th century, wrote the "Aryabhatiya," which contained trigonometric tables and formulas. He introduced the concept of sine and explored its properties. 2. Bhaskara I and Brahmagupta: Mathematicians like Bhaskara I and Brahmagupta made further advancements in trigonometry. They extended trigonometric functions to non-right triangles and introduced new concepts like versine and inverse sine. 3. Influence on Islamic Mathematics: India's trigonometric discoveries had a profound impact on Islamic mathematics during the medieval period. Islamic scholars translated and expanded upon Indian trigonometric works, preserving and disseminating this knowledge.

Islamic mathematics played a crucial role in preserving and advancing trigonometry during the Middle Ages. Three important points about the role of trigonometry in Islamic mathematics: 1. Translation Movement: Islamic scholars translated Greek, Indian, and Babylonian mathematical texts into Arabic. This included works on trigonometry, which were then further developed and refined. 2. Al-Battani and Trigonometric Tables: Al-Battani, also known as Albategnius, made significant contributions to trigonometry in the 9th century. He calculated more accurate values for trigonometric functions and compiled extensive trigonometric tables. 3. Al-Khwarizmi's Influence: Al-Khwarizmi, an influential Islamic mathematician, contributed to the development of trigonometry by refining and standardizing its methods. His work laid the groundwork for the later European development of trigonometry.

Trigonometry was reintroduced to medieval Europe through the translation of Islamic mathematical texts, leading to further developments and applications. Three important points about the development of trigonometry in medieval Europe: 1. Fibonacci's Liber Abaci: Fibonacci, an Italian mathematician, included trigonometric concepts in his book "Liber Abaci" in the 13th century. This helped reintroduce and popularize trigonometry in Europe. 2. Reintroduction of Greek Ideas: European scholars began to revisit and study ancient Greek trigonometric ideas, including chord functions. This led to the formulation of new trigonometric relationships. 3. Regiomontanus and the Renaissance: Regiomontanus, a German mathematician of the 15th century, played a pivotal role in the development of trigonometry during the Renaissance. He improved trigonometric tables and made trigonometry an integral part of astronomy.

Key figures in the history of trigonometry include Hipparchus, who created the first trigonometric table; Ptolemy, known for his work on chords and circles; and the Indian mathematician Aryabhata, who introduced the concept of sine.

Trigonometry continued to develop throughout history, with contributions from Islamic scholars during the Golden Age of Islam. They expanded on Greek and Indian trigonometry and introduced trigonometric functions as we know them today.

Trigonometry played a crucial role in early astronomy, navigation, and surveying. It was used to calculate distances, angles, and positions of celestial objects and landmarks, contributing to the development of science and exploration.

Trigonometry became a fundamental branch of mathematics during the Renaissance and the Scientific Revolution. Mathematicians like Johannes Kepler and Isaac Newton used trigonometric principles to formulate laws of planetary motion and calculus, respectively.

Historical trigonometric theorems and formulas include the Pythagorean theorem, which relates the sides of a right triangle, and the law of sines and law of cosines, which are used to solve non-right triangles.

Modern trigonometry, as we know it today, has evolved over centuries through the contributions of mathematicians from various cultures. Three important points about the evolution of modern trigonometry: 1. Redefining Trigonometric Functions: The transition from chords to the familiar sine, cosine, and tangent functions occurred during the Renaissance. This made trigonometry more versatile and applicable to a wider range of problems. 2. 18th-Century Advances: Mathematicians in the 18th century, such as Leonhard Euler and Jean le Rond d'Alembert, made significant contributions to trigonometry. Euler's work on complex numbers and exponential functions greatly influenced modern trigonometry. 3. Integration with Calculus: Trigonometry became closely integrated with calculus in the 19th century, as mathematicians like Augustin-Louis Cauchy and Joseph Fourier used trigonometric functions to study periodic phenomena and solve differential equations.

Trigonometry remains essential in fields such as physics, engineering, computer graphics, and architecture. It is used in designing structures, analysing waveforms, and solving complex mathematical problems.

To learn more about the history of trigonometry, consider reading books like "A History of Mathematics" by Carl B. Boyer and exploring educational websites and academic journals dedicated to mathematics history.

In conclusion, the history of trigonometry is a fascinating journey that spans millennia and encompasses contributions from diverse cultures and scholars. From its humble beginnings in ancient civilizations to its modern applications in mathematics, science, and engineering, trigonometry has evolved into a fundamental and indispensable branch of mathematics.

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A Note on the History of Trigonometric Functions

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Trigonometric functions appear very frequently in mechanism kinematic equations (for example as soon a revolute joint is involved in the mechanism). Dealing with these functions is di cult and trigonometric substi- tutions are used to transform them into algebraic terms that can be handled more easily. We present briefly the origin of the trigonometric functions and of these substitutions.

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Merlet, JP. (2004). A Note on the History of Trigonometric Functions. In: Ceccarelli, M. (eds) International Symposium on History of Machines and Mechanisms. Springer, Dordrecht. https://doi.org/10.1007/1-4020-2204-2_16

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