Trigonometry before the “sin” key

Trigonometry is the study of angles and triangles. It has many uses: it is used to calculate unknown side lengths and angles of triangles, to measure heights, and even as part of the global positioning system to find positions on the surface of the earth.

Trigonometry has humble beginnings in what the Egyptians called a seked. This was basically a measure of slope which predated the notion of an angle:

In order to give the seked of the line segment AC, it is necessary that the line segment AB be one cubit (arm length, generally the length of the king’s arm). The seked would then be the length of the BC, normally measured in palms and digits.

Later, the Babylonians introduced the notion of an angle in the modern sense of a fraction of the arc around a circle. This stands in contrast to the Egyptian concept of the seked: the Egyptians thought in terms of slope, whereas the Babylonians also entertained the possibility of absolute angle. The Babylonians also introduced the degree to measure angles which is still in use.

The Greeks also made great strides in advancing trigonometry. They were interested in the chord of an angle, designated crd(θ). The chord is more easily drawn than explained:

The chord of an angle. Image: Farmer Jan via Wikimedia Commons

For an angle in a unit circle, the chord is the distance between the points where the angle intersects the circle. The chord is no longer commonly used as a trigonometric function; however, it is related to the sine: crd(θ)=2sin(θ/2).

It’s not simple to calculate trigonometric functions. Before the era of computers, the usual way of knowing trigonometric functions was to look up their values in a table. This is how navigators on ships calculated their positions, how armies knew how to aim catapults, and how many other trigonometric tasks were accomplished. One of the few travesties of the development of computers which perform calculations in the blink of an eye that centuries ago would have taken years is that the genius of those who performed such feats as calculating trigonometric functions is no longer fully appreciated. Ptolemy was one of the first people to publish a table of values of a trigonometric function. He was creative in finding ways to calculate chords of difficult angles. An example of his creativity was his use of the tenth proposition of Euclid’s Elements, which states:

If an equilateral pentagon is inscribed in a circle, then the square on the side of the pentagon equals the sum of the squares on the sides of the hexagon and the decagon inscribed in the same circle.

An equilateral pentagon inscribed in a circle An equilateral hexagon inscribed in a circle An equilateral dodecahedron inscribed in a circle The labeled points of the previous three images

In other words, this proposition states that EA2=DA2+BA2.

This allowed him to calculate the chord of 72°. He was also greatly aided by a simple half angle formula developed earlier by Hipparchus. This only works for chords, but can easily be extended to sines. It states:

crd2(θ/2) = 2-crd(180°-θ)

This half angle formula is incredibly significant. A modern half angle formula called CORDIC is used in calculating sines in computers because it is very efficient computationally. Much of it’s computational work consists of division by two. Because modern computers use binary numbers, they can divide by two very quickly – they only need to shift the numbers by one position.

Another historically significant trigonometric function is the versine. The versine is given by ver(θ)=1-cos(θ). A function called the haversine which is simply half of the versine can be used to calculate great circle distances (the length of the shortest distance between two points) on the surface of the earth or another sphere. Also, the versine is useful because the cosine of very small angles is close to one. This exaggerates rounding errors when one multiplies by the cosine. One solution to this problem is to replace r*cos(θ) with r+r*ver(θ). Notably, Indian astronomers made tables of versine values at some point well before the 12th century CE.

When you enjoy the ability to easily calculate a sine by pressing a button on your calculator, be grateful that you do not have to calculate it manually!

References:

Euclid’s Elements, Book XIII

History of Trigonometry by David E. Joyce of Clark University

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s