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).

## 5 comments:

I agree! I'm always sad when I see books discussing the law of sines without the circumcircle in there.

A couple minor notes: I would say 2R (for circumcircle), not 2r (which to me would be incircle).

Also, your spherical law of sines you tell me is 6V/(abc) or something? I didn't know that one. But if you're going to say it that way, then you should also point out that because Area = (abc)/(4R), then 2R is the same as (abc)/(2*Area) - hm, that looks upside down! Maybe your spherical law of sines should be flipped so that the abc can be on top both times? Or maybe I just made an algebra mistake.

I am always reluctant to disagree with anything that Joshua writes because he has taught me more math than people who were paid to do so... but I think the Area = (abc)/(4R) that he writes is a property of planer triangles and not spherical (I wrote to him to check, and he may yet correct me).

I have corrected an editing mistake in the blog in which I substituted abc for sin(a)sin(b)sin(c) ... quite a difference.

Thanks again Josh for reminding me about appropriate notation for the incircle and circumcircle.

Sorry I wasn't clear enough!

What I'm saying is that we can write, for planar triangles,

a/sin A = etc. = 2R = abc/(2 Area),

and for spherical triangles, I think it turns out that

sin a / sin A = etc. = sin a sin b sin c / (6 V)

In other words, to keep the "a b c" term on top, you should flip your spherical law of sines and have the sin a on top, to make it a more parallel construction to the formula in the planar case.

AHA! Now I get it... and of course, you are right.. thank you

its not necessary to inscribe the triangle in a circle to prove it

u culd create a perpendicular line then there will be no need of the 2R

u will be required to use trigonometric ratio

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