Mathematics - All About Trigonometry (Part 1)

Introduction

Hey it's a me again @drifter1!

Thinking back, quite a few years ago I also wrote some Math articles. A specific category of them caught my intension again: "All About" articles. Like All about Complex numbers for example. Back then I promised to get into Trigonometry as well! Thus, in this article we will make that promise come true, by starting a small series were we will be talking about everything that someone has to know about Trigonometry, Trigonometric Functions, Equations, Formulas, Identities etc. in order to solve problems with triangles (and not only) in Mathematics, Physics and so on. This is the first part that will introduce what (mainly classic) Trigonometry is all about.

So, without further ado, let's dive straight into it!

What is Trigonometry?

Trigonometry is a branch of mathematics that deals with - you guessed it - Triangles. To get more in-depth, in Trigonometry six functions of an angle are the most commonly used. These functions are properties of the angle of a triangle and independent of its size. For ages these values were stored in tables, but now computers made those trigonometry tables obsolete. Using those functions, unknown angles or lengths of geometric figures can be calculated from known or measured angles and lengths.

[Image 2]

Trigonometry was developed, because it was needed for computing angles and distances in fields such as astronomy, mapmaking, surveying and artillery. There are problems that involve angles and distances in two-dimensions, solved by using plane trigonometry, and applications to similar problems for three-dimensional space, which are solved by using spherical trigonometry. In the more modern era, trigonometric functions began to be used in Calculus and Science in general, evolving Trigonometry into Analytic Trigonometry. Trigonometric Functions are nowadays used in almost any branch of science. For example, the extension to nonperiodic functions helped develop quantum mechanics. [Ref 1]

Trigonometry through the Ages

Classic Trigonometry

The word trigonometry comes from the (ancient) Greek words τρĩγονον (triangle) and μέτρον (to measure). Until the 16th century, trigonometry was only about computing missing angles and lengths of a triangle (or any shape that can be dissected into triangles). This is also why we now distinguish trigonometry from geometry, as geometry only focuses on qualitative relations, whilst trigonometry uses quantitative relations. Trigonometry was considered an offspring of geometry and the two became separate branches of mathematics in the 16th century. [Ref 1]

Ancient Egypt and Mediterranean World

Several ancient civilizations possessed a considerable knowledge about practical geometry, which includes some concepts that were a prelude to trigonometry. Using the "run-to-rise" ratio, Egyptians were able to calculate the seked or cotangent of the angle between the base and face of huge constructs like pyramids. Trigonometry in the more modern sense began with the Greeks. In c. 190-120 BCE, Hipparchus was the first to construct a table of values for a trigonometric function. He computed the various parts of a triangle by inscribing it in a circle, so that each side became a chord. To compute the various parts of a triangle, we have to find the length of a chord as a function of the corresponding arc width.

[Custom Figure using GeoGebra]

Using this concept, all geometric terms were defined as relations between the various chords and angles (or arcs). The trigonometric functions were introduced in the 17th century. [Ref 1]

The first major ancient work on trigonometry to reach Europe was the Almagest by Ptolemy in c. 100-170 CE. He lived in Alexandria, which was the intellectual centre of the Hellenistic world, and wrote works on mathematics, geometry and optics. The Almagest was a 13-book compendium on astronomy that became the basis of humankind's world picture until the heliocentric system of Nicolaus Copernicus. Using the so called Babylonian sexagesimal numerals and numeral systems (base 60), which means computing using a circle of radius r = 60 units, so that c = 120 sin (A / 2), where A / 2 is the half-chord. [Ref 1]

[Custom Figure using GeoGebra]

India and Islamic World

The next major contribution to trigonometry came from India. The multiplication or division by 120 in the sexagesimal system was analogous to the multiplication or division by 20 in the decimal system. Thus, Ptolemy's formula can be written as C / 120 = sin B, where B = A / 2 is the half-chord as a function of the arc B that subtends it. This is basically the modern sine function. Using this Aryabhatiya was able to create the first table of sines. The word sinus, which in Arabic was called jiba/jaib, first appeared from the translation of many Greek texts into Latin by Gherardo of Cremona. The abbreviation sin was first used by Edmund Gunter in 1624. [Ref 1]

During the Middle Ages, learning was kept alived by Arab and Jewish scholars in Spain, Mesopotamia and Persia. The first table of tangents and cotangets was constructed in around 860 by Ḥabash Al-Ḥāsib (“the Calculator”). The Arab astronomer Al-Bāttāni (c. 858–929) gave a rule for timekeeping:

where:

θ: elevation of the sun
s: shadow cast by a vertical gnomon
h: height of the gnomon

which is equivalent to the modern formula:

Based on this rule he constructed the first table of cotangents for degrees from 1° to 90°. [Ref 1]

Passage to Europe

Until the 16th centrury, spherical trigonometry was dominating the interest of scholars, which explains the predomincance of astronomy. The first definition of a spherical triangle came from Menelaus of Alexandria (c. 100 CE). A spherical triangle was defined as a figure formed on the surface of a sphere by three arcs of great circles. Several Arab scholars continued to develop spherical trigonometry. Arab scholar Naṣīr al-Dīn al-Ṭūsī was the first to write work on trigonometry that was independent of astronomy (c. 1250). The first modern book entirely about trigonometry appeared in the Bavarian city of Nürnberg in 1533 under the title On Triangles of Every Kind by Regiomontanus. The book contains all the theorems needed to solve triangles, planar or spherical, but only in verbal form, as symbolic algebra was not yet invented at this time. The final major development of classical trigonometry came with the invention of the logarithms by Scottish mathematician John Napier in 1614. [Ref 1]

Modern Trigonometry

In the 16th century trigonometry began to change its character from a purely geometric discipline to an algebraic-analytic subject. Symbolic algebra pioneered by French mathematician François Viète (1540–1603). Analytic geometry by two other Frenchmen, Pierre de Fermat and René Descartes. Viète showed that the solution of many algebraic equations could be expressed using trigonometric expressions. In 1671 James Gregory found the power series for the inverse tangent function (arc tan), demonstrating a connection between π and integers. Later on Isaac Newton (1642-1727) discovered the power series for sine and cosine, which were already known in verbal form by Indian astronomer Madhava (c. 1340-1425). The unification of trigonometry and algebra and the use of complex numbers (x + iy, where i = √(−1)) was completed in the 18th century. Swiss mathematician Leonhard Euler (1707-83) was the one to fully incorporate complex numbers into trigonometry. We all have used Euler's famous formula:

Having trigonometry shift away from its original connection to triangles, the practical aspects were not neglected Quite the opposite happened. Joseph Fourier (1768-1830) discovered that any periodic function could be expressed as a infinite sum of sine and cosine functions. The trigonometric or Fourier series have found numerous applications in almost every branch of science. The extension to nonperiodic functions helped develop quantum mechanics in the early years of the 20th century. [Ref 1]

Triangles

A triangle has three sides and three angles. A simple rule that all triangles follow is that:

The three interior angles always add up to 180° or π radians.

Triangle Naming Convention

There are three special names given to triangles based on how many sides (or angles) are equal:

Equilateral Triangle: Three equal sides and three equal angles (always 60°)
Isosceles Triangle: Two equal sides and angles
Scalene Triangle: No equal sides or angles

Visually:

[Custom Figure using draw.io]

Angle Types

Triangles can also be named by the type of angle:

Acute Triangle: All angles are less than 90°
Right Triangle: Has a right angle (90°)
Obtuse Triangle: Has an angle with more than 90°

Visually:

[Custom Figure using draw.io]

Triangle Perimeter

The perimeter is the distance around the edge of the triangle:

Perimeter = Sum of all three sides

Triangle Area

The area is half of the base times the height:

Visually the base (b) and height (h) are:

[Custom Figure using draw.io]

Triangle Inequality Theorem

Any side of a triangle must be shorter than the other two sides added together.

In other words:

If a side is longer, than the other two sides don't meet up - no triangle
If a side is equal to the other two sides it is not a triangle - but just a straight line

For all triangles with edges a, b and c the following is always true (for it to be a triangle):

Right-Angled Triangle

The triangle with the most interest is the one with a Right Angle. The right angle is commonly shown by a little box in the corner. For another angle, often labeled θ, the three sides get specific names:

Adjacent: adjacent or next to the angle θ
Opposite: opposite the angle θ
Hypotenuse: the longest side

Visually:

[Custom Figure using draw.io]

Dividing the edges of a Right-Angle Triangle with each other, we define the main functions of trigonometry!

RESOURCES:

References

Images

Mathematical equations used in this article, where made using quicklatex.

Final words | Next up

And this is actually it for today's post!

Next time we will start out with Trigonometric Functions...

Also, currently my ideas for "All About" articles include:

Geometry
Polynomial Arithmetic
Exponentials and Logarithms
Rational Expressions

Basically High-School Math Refreshers for Students in Universities or maybe even Middle-Age people that have problems helping their kids out in Math at school!

See ya!

Keep on drifting!