The Physics
Opus in profectus


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triangles, tirangles, tirnagles

Some stuff about triangles.

functions, functions, functions

Some stuff about functions.

Standard position diagram




Reciprocal functions




historical, historical, historical

An arc of a circle looks like a bow (as in a bow and arrow). The geometric word arc is related to the military word archer — a person who shoots arrows with a bow. A straight line joining the ends of an arc looks like a bowstring. Another word for a string is a cord — a word with multiple meanings that used to be spelled differently. The new spelling of cord (without an h) refers to a string or rope and the old spelling of chord (with an h) refers to the straight line joining the ends of an arc. (The musical chord has a completely different origin.) The Greek astronomer Hipparchus of Nicaea usually gets credit for inventing this term, along with the rest of trigonometry, sometime in the 2nd century BCE. The Greek word for string is χορδη (khorde).

Sanskrit is a four thousand year old language that was once the common tongue of scholars in India, Pakistan, Bangladesh, Sri Lanka, Nepal, and beyond. Sanskrit is to the Indian subcontinent as Latin is to the European continent. Both languages were used to spread ideas and information across large geographic regions with great linguistic diversity. Both languages are often described as "dead". The only scholars still using Latin or Sanskrit are those that study original historical texts. No new scholarship is written in Sanskrit or Latin, but some is written about Sanskrit or Latin. Languages never really die, however, as dead languages are the foundation for living ones.

The diagram below shows a variation of the standard position triangle that first appeared in Sanskrit astronomical texts some time before the 12th century. The most famous of these is the सूर्यसिद्धान्त (Surya Siddhanta) or Theory of the Solar System.

Geometric drawing labeled in Sanskrit and English

The Sanskrit word for bow is चाप (ca̅pa). It is also the name given to an arc of a circle. The Sanskrit word for bowstring is ज्या (jya̅). It is also the name given to a chord of a circle. At some point, Indian astronomers found that knowing the size of half a chord was more useful than knowing the size of a whole chord. Half a chord in Sanskrit is ज्या अर्ध (jya̅ ardh). This term became so popular that the modifier अर्ध (ardh) was dropped and the the word ज्या (jya̅) or the similar word जीव (ji̅va) came to mean half a chord all by itself.

Arab scholars transliterated जीव (ji̅va) to جيب (jiba) adding a loan word to Arabic. European scholars didn't know this, so they read it as an already existing one. Since vowels are not really written in Arabic, the letters جيب might represent jiba (a loan word) or they might represent jaib (a preexisting word). The Latin scholars went with the latter. The Arabic word جيب (jaib) translates into English as bosom and Latin as sinus. English scholars did not repeat this mistake. The Latin word sinus was added to English as the loan word sine, not as the translated word bosom.

The origin of the English word sine
language word translation
Sanskrit ज्या अर्ध (jya̅ ardh) bowstring half
ज्या (jya̅) bowstring
जीव (ji̅va) bowstring
Arabic جيب (jiba) [loan word]
جيب (jaib) bosom
Latin sinus bosom
English sine [loan word]

In the old Sanskrit text, half a chord was called ज्या (jya̅). The line that cut the chord in half was called कोटि ज्या (koṭi-jya̅) on one side and उत्क्रम ज्या (utkrama-jya̅) on the other. You could think of them as "complimentary chord" and "contrary chord". When ज्या (jya̅) became sinus became sine, कोटि ज्या (koṭi-jya) or कोज्या (kojya) became cosinus became cosine. Sine and cosine have the same value for complementary angles. This concept was reused for the other cofunction pairs — tangent has cotangent, secant has cosecant.

The leftover piece on the diagram, उत्क्रम ज्या (utkrama-jya̅), became versus sinus in Latin and versed sine or versine in English. The versed part is related to the Latin word versus in the sense of being against something, but I don't really get it. There's a whole extended family of co-, ver-, cover-, ha-, haver-, cohaver-, hacover-, and ex- functions that aren't used much anymore. I will not be discussing them.

That leaves us with tangent and secant. The diagram below shows another variation of the standard position triangle with three extra line segments that form a triangle outside the circle — one that starts at the center of the circle and cuts across the arc like an arrow waiting to be fired, a second that leans against the circle touching it only at the point where the radius ends, and a third at right angles to the first that also cuts across the circle.

Geometric drawing labeled in English

The line segment touching the circle gives us the tangent and cotangent pieces (one to the right of and one to the left of the point touching the circle). The English word tangent comes from the Latin word tangens — touching. The line segments cutting across the circle give us the secant and cosecant pieces (one cutting the circle horizontally and the other cutting the circle vertically). The English word secant come from the Latin word secans — cutting. The words tangens and secans were given their mathematical meaning by the Danish mathematicianThomas Finckein 1583. Earlier Arab, Hindu, Roman, Greek, and Babylonian mathematicians probably knew of these concepts, but their words do not seem to have made it into many modern languages (Modern Greek being a major exception).

special triangles

Standardized tests love special triangles. One reason may be to help students solve problems quickly, especially if they aren't allowed to use calculators. The real reason is probably to speed up grading. Instructors can spot math and reasoning errors quicker. Memorizing the ratio of the sides and values of the angles in these triangles is often recommended.

45-45-90 triangle
An isosceles right triangle. The base angles are both 45°. The two legs are equal in size. If we assume each leg has a length of 1 then, according to Pythagorean theorem, the hypotenuse has a length of √2.
30-60-90 triangle
Half an equilateral triangle. Every equilateral triangle has three 60° angles. Assume the sides of the full triangle have a length of 2. Split the whole thing in half with a perpendicular bisector. This gives us one bisected angle of 30° and one bisected side with a length of 1. Using Pythagorean theorem, the leg that was the perpendicular bisector has a length of √3 and is opposite the 60° angle.
3-4-5 triangle
The simplest Pythagorean triple gives us a triangle that's easy to work with. The math of the sides is simple. The size of the angles are best determined with a calculator.