# Who are credited for angle transformation formulae and law of sines in trignometry

I'd like to who are credited for discovering angle transformation formulae $$\sin(A\pm B)=\sin(A)\cos( B)\pm\cos(A)\sin(B)$$ $$\cos(A\pm B)=\cos(A)\cos( B)\mp\sin(A)\sin(B)$$ $$\tan(A\pm B)=\frac{\tan(A)\pm\tan(B)}{1\mp\tan(A)\tan(B)}$$ $$\cot(A\pm B)=\frac{\cot(A)\cot(B)\mp1}{\cot(A)\pm\cot(B)}$$ and so on, as well as and law of sines $$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$$ in trigonometry.

• Menelaus (c. 100 AD) gives a spherical law of sines in his Sphaerica, which suggests that the plane one was already known, possibly, to Hipparchus (c.150 BC). Angle sum formulas are usually credited to the medieval Persian mathematician al-Būzjānī, see Wikipedia's History of trigonometry. – Conifold Apr 5 '19 at 21:10

Your first three formulas (addition laws) were essentially known since the beginning of trigonometry, though they were not written in the modern form. These laws actually predate the definition of $$\sin$$, $$\cos$$ and $$\tan$$. The only trigonometric function in the ancient Greece was the chord $${\mathrm{chd}}$$ which is related to our modern $$\sin$$ as follows $$\mathrm{chd}\, x=2\sin(x/2)$$. Ptolemy, who computed the first trigonometric table makes use of the addition formula for the chord, which is equivalent to Menelaus geometric theorem. Later, when other trigonometric functions were introduced (for convenience of computation) the addition laws were of course written for them as well. It is a routine matter to rewrite the addition formula for any trigonometric function, once you know it for one of them.

Your third formula, the sine law for a flat triangle, is the limit case of the similar formula for the spherical triangle, $$\frac{\sin a}{\sin A}=\frac{\sin b}{\sin B}=\frac{\sin c}{\sin C}.$$ When the triangle is small, the sines of the sides are approximately equal to the sides themselves. Spherical trigonometry actually predates the flat one. Wikipedia article "Law of sines" credits it to various medieval astronomers in the Middle East, and in Europe - to Johannes Müller (Regiomontanus).

Reference. Glen Van Brummelen, in his book Heavenly Mathematics, Princeton 2013 credits the spherical law of sines to Abu Nasr and Abu 'l-Wafa, and their priority dispute is discussed by Al-Biruni. (The special case of the flat Law of sines was stated much later).

The answer to the question seems to depend in part on how far the questioner is prepared to accept translations of the modern formulae in terms of earlier mathematics.

Here is a part-answer relating to the sine theorem, showing its effective equivalence to a result of Euclid.

The modern sine theorem appears only to restate in limited terms a core fact proved by Euclid (c. 300 BC) -- but only translated across different languages and cultures.

Thus, Euclid's Elements III.20-21, showed that the angle subtended at the center of a circle by any chord is twice the angle subtended by the same chord at any point of the circumference.

Where the chord is a, and the angle subtended at the cirumference is A, then the angle at the center is 2A. By only translating into modern terms, and without proving any further fact,

    a =  chord(2A) ,


(by the result of Euclid and the definition or construction of the chord).

But the modern sine of A, and the ancient chord of angle 2A inscribed in a circle of radius R, are related by definition or by construction as

    sin(A) = 1/2R chord(2A) ,


so it only appears to say the same thing as Euclid III.20-21, but in a translated and rearranged way, to put

    a / sin(A)  = 2R (a constant),


with the same relation clearly applying also where the side and opposite angle are b and B, or c and C (or even any x and X, where x is any chord whatever of the same circle with radius R, and X is any of the angles it subtends on the circumference). The modern sine theorem is clearly included within the generality of that reformulation.

• At least a comment explaining reason for the markdown made here would be generally helpful. – terry-s Apr 6 '19 at 9:51
• I have no idea about what might have been the reason for a down-vote, but your first sentence alone gets a point from me. That single sentence, when suitably modified, probably applies to most every question that gets asked here. Related are the Historiographic vices papers I give freely available links to in this comment. – Dave L Renfro Apr 6 '19 at 11:37
• Thanks very much for the links to the really useful Historiographic vices papers! -- and for your comment. – terry-s Apr 6 '19 at 14:24