Time, and Trig, and Conics, Oh MY!

A while back I wrote about the equation of time, recalling in part,

It came from a lesson in trig on simple harmonic motion. We were talking about things that demonstrated sinusoidal behavior, and one bright young man suggested that the height of the sun at noon would be an example. I sort of agreed with a comment about "not exactly at noon.. but" and then the little guy was confused.. "You know, I said, like today ."(it was Feb 12) "I think the sun was about 12 minutes late or so."
Slow looks at each other, then back to me... the three letter word look,,,,,"Huh?"
"You know, that's what the analemma is for, telling if the sun is early or late.".....

Same look, compounded by the wild eye..."HUH?"

See the whole blog here

So anyway, today I opened the blog over at "Square Circle Z" by Zac,(who seems to prefer to be called, "the mysterious Zac") and it showed a really nice use of the addition of two trig functions to explain the two components of the Equation of time. A nice exercise for people who tend to patienly explain to students who ask "What's that good for?"...(my personal first instinct is to throw things, but many administrations frown on that)

Don't Write the Law of Sines Upside Down, Please!

Mostly I'm a teach and let teach kind of guy... you do what you think is important, and let me do the same... but sometimes when I see people teach the law of sines... I wonder... DO THEY KNOW?????
The ideas behind the law of sines, like those of the law of cosines, predate the word sine by over a thousand years. Theorems in Euclid on lengths of chords are essentially the same ideas we now call the law of sines. The law of sines for plane triangles was known to Ptolemy and by the tenth century Abu'l Wefa had clearly expounded the spherical law of sines (in 2014 Thony Christie sent a note telling me that "Glen van Brummelen in his "Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry" says the spherical law of sines was discovered either by Abū al-Wafā or Abu Nasr Mansur .  It seems that the term "law of sines" was applied sometime near 1850, but I am unsure of the origin of the phrase (and if you have a reference, please advise).



A simple proof of the law of sines begins with a triangle, ABC, inscribed in a circle with radius R. A diameter is drawn with one endpoint at A terminating at D and the right triangle ADC is created. Using the right triangle definitions of Sine, we see that sin (ADC)=AC/AD.

Because Angles ABC and ADC are both inscribed angles cutting the same arc, they have equal measures, and therefore equal sines. By substitution then we get sin(B)=AC/AD and since AD is a diameter equal to 2R , we may also write sin(B) =AC/2R . Now if we adopt the modern convention of calling the side AC opposite angle B, side b, we can rewrite this as sin(B)= b/2R. With one last algebraic manipulation we exchange the positions of sin(B) and 2R to get 2R= b/sin(B) [Thanks to Joshua Zucker who reminded me that the radius of the circumcircle is usually capitalized, with r used for the radius of the incircle]. Since the choice of angle B was arbitrary, we could show that the same holds for each side and opposite angle pair, producing the typical high school textbook theorem below. 


(as per Joshua Zucker's note below, that r should probably be R since that is more commonly used for the radius of the Circumcircle, with r used for the incircle. I leave it rather than having to scrabble together a new graphic)
Addendum: [ I still get emails and comments that seem not to realize that by writing the sides on top you can reduce the entire thing to a single geometric relationship.  For any triangle, the ratio of a side to the sine of the opposite angle is always equal to the diameter of the circle which circumscribes the triangle.  Perhaps I should have added this earlier.]

I am frequently amazed to see this theorem presented in math texts without the "=2R" which seems to give it visual or geometric life. It is especially curious since the property dates back to Ptolemy. I get even more frustrated when it is presented with the Angles on top, thus destroying the geometric meaning. I can't think of a good reason for doing that, but if you consciously do it the other way for some reason, I would love to hear it.

As a footnote, in spherical triangles it is customary to work with a sphere of unit radius, thus allowing the sides to be expressed in radian or angle measure as well as the angles. Since all great circles have length 360 degrees, we may express the length of a side by the fraction of a complete great circle it occupies. With this convention, the spherical law of sines states that in a spherical triangle with sides a, b, and c and angles A, B, and C, it is true that

\( \frac{sin a}{sin A} = \frac{sin b}{sin B} = \frac{sin c}{sin C}  = \frac{sin a sin b sin c}{6 Vol(OABC)} \)



That is the ratio of  the sin of any side to the sin of its opposite angle  the product of the sines of the sides over six times the volume of the tetrahedron formed by the center of the sphere and the points A, B, and C.

According to Ubiratàn D'Ambrosio and Helaine Selin, the spherical law of sines was discovered in the 10th century. It is variously attributed to al-Khujandi, Abul Wafa Bozjani, Nasir al-Din al-Tusi and Abu Nasr Mansur. 
Ptolemy knew the formula  for the planer law of sines and something like the angle addition formula but he expressed them in terms of chords of arcs, not sines of angle.  The half chords, or sines, were introduced by the Hindu mathematician Aryabhata around 500.
The spherical law of sines was first presented in the west by Johann Muller, also known as Regiomontus,in his De Triangulis Omnimodis in 1464. This was the first book devoted wholly to trigonometry (a word not then invented). David E. Smith suggests that the theorem was Muller's invention. The word trigonometry, by the way, seems to have been the creation of Bartholomaus Pitiscus, who used it in the title of a book, Trigonometriae sive de dimensions triangulorum libri cinque in 1595. Among other things the book includes a demonstration of the law of sines and the law of cosines. I find it highly unusual that the first use of a word would be in the title of a book.

As a second footnote, it may be of interest to teachers and students that the use of the unit circle was "unknown much before 1800". I found that out in an article on "Benjamin Banneker's Trigonometry Puzzle" by Florence Fasanelli, Graham Jagger, and Bea Lumpkin that appeared in the MAA online magazine Convergence. Unfortunatly the magazine is no longer free on-line. Older trig tables gave the measurements for the sine, tangent and secant on a circle of very large radius (van Schooten used 10,000,000) rather than on a circle of radius 1, as we do today. Thus, the sin 90°, also called the “total sine” was given as 10,000,000, and the sine of 45° was 707,107 and not 0.707107, as we would use today. Anyone using these tables would use rules of proportion to make any necessary conversions.

Next I hope to talk about Descartes Rule of Signs.. (which has almost completely disappeared from my current textbooks).

Rheticus, and The Names of Trigonometric Ratios

Spending lots of time lately reading old English journal articles (1825-45) sent me by Dave Renfro who trys to help me stay up on the history of math. It is kind of great reading and watching the actual history of ideas unfold as they did in the old journals.... I came across an interesting letter from Agustus De Morgan about the protege of Copernicus, George Joachim of Rhaetia, also called Rheticus. It was Rheticus who managed to convince Copernicus to publish his long withheld manuscript.
I didn't realize for some years of teaching that in the early days the trigonometric functions were conceived to be lengths of segments in a circle of a given diameter (or radius) rather than the more modern view of ratios. I did not know until I read this article, that apparently it was Rheticus who first developed this approach. In fact, the tables he created to include in his publication of the trigonometric sections of De Revolutionibus were the first tables to include cosines (although he did not use these names). Here is the way De Morgan wrote it:
" Modern teachers (he writes in 1845) of trigonometry have pretty generally abandoned the system of independent lines, which used to be called sines, tangents, &c.; and have substituted, for the meaning of these words, the ratio of the sides of right-angled triangles. It appears that they have antiquity in their favor; indeed so completely has the idea of representing the ratios of the sides of triangles taken possession of the mind of Rheticus, that he abandons the use of the word sine. He dwells on the importance of the right-angled triangles, without any reference to the circle: his maxim expressed in the dialogue, is Triquetrum in planicie cum angulo recto, est magister Mathesos . It would also seem as if his choice of the semi-quadrantal arrangement with double descriptions was dictated merely by the convenience of heading one division with majus latus, and the other with minus latus. [Rheticus had labeled the top of his table with perpendiculum and basis, then the bottoms of these columns were reversed, much as Sine and Cosine were reversed at the top and bottom of tables used in my youth before calculators]........ The names cosine, cotangent, and cosecant are the consequence, not the cause, of this duplicate system of arrangements.......The introduction of the terms sine of the complement, complemental sine, and cosine, &c., followed after an interval of more than half a century."
De Morgan points out that one of the reasons it is so hard to find copies of much of Rheticus' work is that ", In the Index Expurgatorius, it is not Copernicus who is forbidden to be read generally; the prohibition only extends to the work De Revolutionibus, and is accompanied with a nisi corrigatur. But Rheticus is wholly forbidden to be read in any of his works. "
I think the difference in the two mens treatment in the Index may be because of the fact that Rheticus was Protestant, and in fact, was at Wittenburg, the very University where Luther had taught, and burned the Papal Bull.

An interesting anecdote told about Rheticus while he was, "puzzling himself about the motion of Mars, he invoked his genius or guardian angel to help him out of the difficulty: the angel accordingly lifted him up by the hair of his head to the roof and threw him down upon the pavement saying with a bitter laugh, 'That's the way Mars moves.' "

addendum James asked about the phrase "semi-quadrantal"... this just means he only went from 0 degrees to 45 degrees (1/2 of a quadrant) and then put Sin-Cos (he didn't use these words) at the top of the columns and Cos-Sin in the reverse order at the bottom... so that from 45 to 90 degrees was simply read up from the bottom....My old CRC tables were arranged the same way, and many textbooks did as well in the Fifties-sixties
 Giving Sin and Cosine as ratios was still pretty new when this was written by De Morgan. It appears that Peacock had initiated the practice in his lectures at Cambridge around 1830, and by 1837, according to De Morgan, it had become the accepted way to define the terms.