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) . 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. 

\(\frac{a}{Sin A} =\frac{b}{Sin B} =\frac{c}{Sin C} = 2R\)

I am frequently disappointed to see this theorem presented in math texts without the "=2R" which seems to give it visual or geometric life, and even more so when the angles are in the numerator.... not wrong, but just not beautiful. It is especially curious since the property dates back to Ptolemy. 

Since the area of triangle ABC can be written as Area= a*b*c/4R ,  then \(2R = \frac {abc}{2 * Area }\)

I especially like this suggested approach from Joshua Zucker because it relates to a similar relation in spherical geometry

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

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

That is the ratio of  the sine of any side to the sine of its opposite angle is 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. Unfortunately 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.