Law of Sines

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. It seems that the term "law of sines" was applied sometime near 1850, but I am unsure of the origin of the phrase.

 

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 ABD is created. Using the right triangle definitions of Sine, we see that sin (ADB)=AB/AD. Because ACB and ADB are both inscribed angles cutting the same arc, they have equal measures, and therefore equal sines. By substitution then we get sin(C)=AB/AD and since AD is a diameter equal to 2r , we may also write sin(C) =AB/2r . Now if we adopt the modern convention of calling the side AB opposite angle C, side c, we can rewrite this as sin(C)= c/2r. With one last algebraic manipulation we exchange the positions of sin(C) and 2r to get 2r= c/sin(C). Since the choice of angle C was arbitrary, we could show that the same holds for each side and opposite angle pair, producing the typical high school textbook theorem below.

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.

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

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 The spherical law of sines was first presented 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.