The world of trigonometry

Trigonometry is one of the oldest and widely used branches of mathematics. It is interesting how many cultures came to the same conclusions and how many parallels can be drawn between their notes. The study of trigonometry started around 2nd millennium BC in Egypt and Mesopotamia.

Firstly, trigonometry wasn’t considered math at all, but a calculation tool. The trigonometry we know now didn’t show until 18th century. Egyptians developed trigonometry from measuring land and building pyramids.

While Egyptians were builders, Mesopotamians were astronomers and mostly developed their knowledge in trigonometry trying to calculate distance and mapping the stars and planets.

We can find trigonometry in almost every aspect of our lives, which is why it isn’t that hard to picture it and learn it. First thing you should learn is some basic trigonometric ratios in a right triangle.

You’ll learn about sine, cosine, tangent and cotangent and some of their uses. This introduction will lay some grounds for you to learn more and more.

Even after one small lesson you’ll be able to calculate so many real life problems with little effort.

Whether you are a beginner and want to start learning about trigonometry, or you simply want to learn more, we have what you need!

The World of Trigonometry

Trigonometric identities and examples

The main Pythagorean identity is the notation of Pythagorean Theorem in made in terms of unit circle, and a specific angle.

The World of Trigonometry

Methods of solving trigonometric equations and inequalities

Trigonometric equations are equations that come in form f(x) = a where a is a real number, and f(x) some trigonometric function.

The World of Trigonometry

Unit circle definition of trigonometric functions

What is unit circle, where is unit of trigonometric function placed on circle and find special angles in trig.

The World of Trigonometry

Basic trigonometric functions

Trigonometry is important part of geometry that includes calculation of sine, cosine, tangent, cotangent, etc.

Proofs made easy

Pythagoras was ancient Greek mathematician. He made an incredible discovery about triangles. The legend has it that one day he was very, very bored and as he stared at the tiles on the floor he started noticing a curious pattern. The tiles were in the shape of isosceles triangles.

Here he noticed a very important relation between sides of a triangle. He came to his famous Pythagorean Theorem. This idea evolved, and after some time, the great Pythagorean Theorem got a new notation which is called the main Pythagorean identity. Soon enough, two more identities developed.

People recognized how important this is and kept on researching.

As people researched, many new formulas accrued. Now we know formulas for angle addition, law of cosines and law of sines, many trigonometric identities and product and sum identities. This all makes proving trigonometric theorems very easy because we can use so many statements we know are true. You can learn everything about this here.

This is how, from one boring day and extraordinary mind, one great branch of geometry was born.

Looking at things from a different angle

When you already get aquatinted with angles and trigonometric functions we’ll show you what they actually represent on a unit circle. You will learn another way of looking at angles and how to find them if you know only their sine, cosine, tangent or cotangent value, what is the easiest way to draw them and how to memorize things you’ll need.

Don’t worry if this sounds a bit complicated, we provided you with many examples you can follow so you can do everything on your own soon enough. If some things are hard for you to learn by heart, we’ll always find a way to make it easier for you. You’ll find out what are the most common special angles and why are they so important to us.

So, get your compass and pencil in your hands, open the lessons and start learning.

Missing equations and inequalities?

Now to worry, there are plenty tasks and problems in trigonometry that include equations and inequalities. Don’t start panicking, we got you covered. Everything you need you can find in these lessons.

We won’t lie, this part isn’t that easy and it requires work, but with a little bit of your time and dedication you’ll have no problem with mastering this field. There are more than one type of trigonometric equation, and there are more than one way to solve them.

We solved and explained every type of a problem you may encounter in your trigonometric journey.

Why is trigonometry so important?

History of Trigonometry – Part 3

This is the third of three articles on the History of Trigonometry.

Part 2 (Sections 5 – 7) can be found here. Part 1 (Sections 1 – 4) can be found here.

8. The Arabs collect knowledge from the known world

The Arab civilisation traditionally marks its beginning from the year 622 CE the date when Muhammad, threatened with assassination, fled from Mecca to Medina where Muhammad and his followers found safety and respect.

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Over a century later, the Arabs had established themselves as a powerful unified force across large parts of the Middle East and The Caliph Abu
Ja'far Al-Mansour moved from Damascus to establish the city of Baghdad during the years 762 to 766.

Al-Mansour sent his emissaries to search for and collect knowledge. From China, they learnt how to produce paper, and using this new skill they started a programme of translation of texts on mathematics, astronomy, science and philosophy into Arabic.

This work was continued by his successors,
Caliphs Mohammad Al-Mahdi and Haroun Al-Rasheed. The quest for knowledge became a lasting and significant part of Arab culture.

Al-Mansour had founded a scientific academy that became called 'The House of Wisdom'.

This academy attracted scholars from many different countries and religions to Baghdad to work together and establish the traditions of Arabic science that were to continue well into the Middle Ages.

Some of this work was later translated into Latin by Mediaeval scholars and passed on into Europe. The
dominance of Baghdad and the influence of the Arab World was to last for the next 500 years.

The scholars in the House of Wisdom came from many cultures and translated the works of Egyptian, Babylonian, Greek, Indian and Chinese astronomers and mathematicians. The Mathematical Treatise of Ptolemy was one of the first to be translated from the Greek into Arabic by Ishaq ben Hunayn (830-910).

It was admired for its extensive content and became known
in Arabic as Al-Megiste (the Great Book). The name 'Almagest' has continued to this day and it is recognized as both the great synthesis and the culmination of mathematical astronomy of the ancient Greek world.

It was translated into Arabic at least five times and constituted the basis of the mathematical astronomy carried out
in the Islamic world.

9. India: The Sine, Cosine and Versine

What Is Trigonometry?

Trigonometry is a branch of mathematics that studies relationships between the sides and angles of triangles.

Trigonometry is found all throughout geometry, as every straight-sided shape may be broken into as a collection of triangles.

Further still, trigonometry has astoundingly intricate relationships to other branches of mathematics, in particular complex numbers, infinite series, logarithms and calculus. 

The word trigonometry is a 16th-century Latin derivative from the Greek words for triangle (trigōnon) and measure (metron). Though the field emerged in Greece during the third century B.C., some of the most important contributions (such as the sine function) came from India in the fifth century A.D.

Because early trigonometric works of Ancient Greece have been lost, it is not known whether Indian scholars developed trigonometry independently or after Greek influence.

According to Victor Katz in “A History of Mathematics (3rd Edition)” (Pearson, 2008), trigonometry developed primarily from the needs of Greek and Indian astronomers.

An example: Height of a sailboat mast

Suppose you need to know the height of a sailboat mast, but are unable to climb it to measure.

If the mast is perpendicular to the deck and top of the mast is rigged to the deck, then the mast, deck and rigging rope form a right triangle.

If we know how far the rope is rigged from the mast, and the slant at which the rope meets the deck, then all we need to determine the mast’s height is trigonometry.

For this demonstration, we need to examine a couple ways of describing “slant.” First is slope, which is a ratio that compares how many units a line increases vertically (its rise) compared to how many units it increases horizontally (its run).

Slope is therefore calculated as rise divided by run. Suppose we measure the rigging point as 30 feet (9.1 meters) from the base of the mast (the run). By multiplying the run by the slope, we would get the rise — the mast height. Unfortunately, we don’t know the slope.

We can, however, find the angle of the rigging rope, and use it to find the slope. An angle is some portion of a full circle, which is defined as having 360 degrees. This is easily measured with a protractor.

Let’s suppose the angle between the rigging rope and the deck is 71/360 of a circle, or 71 degrees.

We want the slope, but all we have is the angle. What we need is a relationship that relates the two. This relationship is known as the “tangent function,” written as tan(x). The tangent of an angle gives its slope. For our demo, the equation is:  tan(71°) = 2.90. (We'll explain how we got that answer later.)

This means the slope of our rigging rope is 2.90. Since the rigging point is 30 feet from the base of the mast, the mast must be 2.90 × 30 feet, or 87 feet tall. (It works the same in the metric system: 2.90 x 9.1 meters = 26.4 meters.)

Sine, cosine and tangent

Depending on what is known about various side lengths and angles of a right triangle, there are two other trigonometric functions that may be more useful: the “sine function” written as sin(x), and the “cosine function” written as cos(x). Before we explain those functions, some additional terminology is needed.

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Sides and angles that touch are described as adjacent. Every side has two adjacent angles. Sides and angles that don’t touch are described as opposite. For a right triangle, the side opposite to the right angle is called the hypotenuse (from Greek for “stretching under”).

The two remaining sides are called legs.

Written in stone: the world’s first trigonometry revealed in an ancient Babylonian tablet

The ancient Babylonians – who lived from about 4,000BCE in what is now Iraq – had a long forgotten understanding of right-angled triangles that was much simpler and more accurate than the conventional trigonometry we are taught in schools.

Our new research, published in Historia Mathematica, shows that the Babylonians were able to construct a trigonometric table using only the exact ratios of sides of a right-angled triangle. This is a completely different form of trigonometry that does not need the familiar modern concept of angles.

At school we are told that the shape of a right-angled triangle depends upon the other two angles. The angle is related to the circumference of a circle, which is divided into 360 parts or degrees. This angle is then used to describe the ratios of the sides of the right-angled triangle through sin, cos and tan.

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But circles and right-angled triangles are very different, and the price of having simple values for the angle is borne by the ratios, which are very complicated and must be approximated.

The three ratios of a modern trigonometric table, rounded to six decimal places, with auxiliary angle Θ in degrees. Daniel Mansfield, Author provided

This approach can be traced back to the Greek astronomer and mathematician Hipparchus of Nicaea (who died after 127 BCE). He is said to be the father of trigonometry because he used his table of chords to calculate orbits of the Moon and Sun.

But our new research shows this was not the first, or only, or best approach to trigonometry.

Babylonian trigonometry

The Babylonians discovered their own unique form of trigonometry during the Old Babylonian period (1900-1600BCE), more than 1,500 years earlier than the Greek form.

Remarkably, their trigonometry contains none of the hallmarks of our modern trigonometry – it does not use angles and it does not use approximation.

The Babylonians had a completely different conceptualisation of a right triangle. They saw it as half of a rectangle, and due to their sophisticated sexagesimal (base 60) number system they were able to construct a wide variety of right triangles using only exact ratios.

The Greek (left) and Babylonian (right) conceptualisation of a right triangle. Notably the Babylonians did not use angles to describe a right triangle. Daniel Mansfield, Author provided

The sexagesimal system is better suited for exact calculation. For example, if you divide one hour by three then you get exactly 20 minutes. But if you divide one dollar by three then you get 33 cents, with 1 cent left over. The fundamental difference is the convention to treat hours and dollars in different number systems: time is sexagesimal and dollars are decimal.

The Babylonians knew that their sexagesmial number system allowed for many more exact divisions. For a more sophisticated example, 1 hour divided by 48 is 1 minute and 15 seconds.

This precise arithmetic of the Babylonians also influenced their geometry, which they preferred to be exact. They were able to generate a wide variety of right-angled triangles within exact ratios b/l and d/l, where b, l and d are the short side, long side and diagonal of a rectangle.

The ratio b/l was particularly important to the ancient Babylonians and Egyptians because they used this ratio to measure steepness.

The Plimpton 322 tablet

We now know that the Babylonians studied trigonometry because we have a fragment of a one of their trigonometric tables.

Plimpton 322 is a broken clay tablet from the ancient city of Larsa, which was located near Tell as-Senkereh in modern day Iraq. The tablet was written between 1822-1762BCE.

In the 1920s the archaeologist, academic and adventurer Edgar J Banks sold the tablet to the American publisher and philanthropist George Arthur Plimpton.

Plimpton bequeathed his entire collection of mathematical artefacts to Columbia University in 1936, and it resides there today in the Rare Book and Manuscript Library. It’s available online through the Cuneiform Digital Library Initiative.

In 1945 the tablet was revealed to contain a highly sophisticated sequence of integer numbers that satisfy the Pythagorean equation a2+b2=c2, known as Pythagorean triples.

This is the fundamental relationship of the three sides of a right-angled triangle, and this discovery proved that the Babylonians knew this relationship more than 1,000 years before the Greek mathematician Pythagoras was born.

See also:  Possessives

The fundamental relation between the side lengths of a right triangle. In modern times this is called Pythagoras’ theorem, but it was known to the Babylonians more than 1,000 years before Pythagoras was born.

Plimpton 322 has ruled space on the reverse which indicates that additional rows were intended. In 1964, the Yale based science historian Derek J de Solla Price discovered the pattern behind the complex sequence of Pythagorean triples and we now know that it was originally intended to contain 38 rows in total.

The other side of the Plimpton 322 tablet. UNSW/Andrew Kelly, Author provided

The tablet also has missing columns, and in 1981 the Swedish mathematics historian Jöran Friberg conjectured that the missing columns should be the ratios b/l and d/l. But the tablet’s purpose remained elusive.

The first five rows of Plimpton 322, with reconstructed columns and numbers written in decimal.

The surviving fragment of Plimpton 322 starts with the Pythagorean triple 119, 120, 169. The next triple is 3367, 3456, 4825. This makes sense when you realise that the first triple is almost a square (which is an extreme kind of rectangle), and the next is slightly flatter. In fact the right-angled triangles are slowly but steadily getting flatter throughout the entire sequence.

Watch the triangles change shape as we go down the list.

So the trigonometric nature of this table is suggested by the information on the surviving fragment alone, but it is even more apparent from the reconstructed tablet.

This argument must be made carefully because modern notions such as angle were not present at the time Plimpton 322 was written. How then might it be a trigonometric table?

Fundamentally a trigonometric table must describe three ratios of a right triangle. So we throw away sin and cos and instead start with the ratios b/l and d/l. The ratio which replaces tan would then be b/d or d/b, but neither can be expressed exactly in sexagesimal.

Instead, information about this ratio is split into three columns of exact numbers. A squared index and simplified values of b and d to help the scribe make their own approximation to b/d or d/b.

No approximation

The most remarkable aspect of Babylonian trigonometry is its precision. Babylonian trigonometry is exact, whereas we are accustomed to approximate trigonometry.

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The Babylonian approach is also much simpler because it only uses exact ratios. There are no irrational numbers and no angles, and this means that there is also no sin, cos or tan or approximation.

Trigonometry/A Brief History of Trigonometry

A painting of the famous greek geometrist, and “father of measurement”, Euclid. In the times of the greeks, trigonometry and geometry were important mathematical principles used in building, agriculture and education.

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]

First Tables of Sines or Cosines[edit]

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[edit]

Pythagoras, depicted on a 3rd-century coin
In a right triangle: The square of the hypotenuse is equal to the sum of the squares of the other two sides.

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]

The area A {displaystyle A}
of a triangle whose sides have lengths a , b , c {displaystyle a,b,c}
is

A = s ( s − a ) ( s − b ) ( s − c ) {displaystyle A={sqrt {s(s-a)(s-b)(s-c)}}}

where s {displaystyle s}
is the semiperimeter of the triangle:

s = a + b + c 2 {displaystyle s={frac {a+b+c}{2}}}

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

A = ( s − a ) ( s − b ) ( s − c ) ( s − c ) ( s − d ) {displaystyle A={sqrt {(s-a)(s-b)(s-c)(s-c)(s-d)}}}

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).

References[edit]

  1. ↑ http://www.lscc.edu/faculty/steven_a_boast/Shared%20Documents/MAC%201114%20Trigonometry/Activities%20and%20Labs/MAC%201114%20Measuring%20Angles%20Lab.pdf
  2. Complete Dictionary of Scientific Biography. 2008. 
  3. ↑ “Hipparchus of Rhodes”. 1999. http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Hipparchus.html. 
  4. ↑ George Johnston Allman (1889). Greek Geometry from Thales to Euclid

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