1 In general, I have in mind these traditional divisions: Ancient Greek, Indian, Chinese, Arabic-Islamic, and Renaissance Europe (modern times). In reality, the trigonometric functions in the three mentioned stages could be three completely different concepts, but I will connect them together following the principle of history: things start at a point and then develop, forward and sometimes backward, as exactly as in the case of the trigonometric functions: first they dealt with arcs, and then angles, and then back to deal again with arcs (i.e. real and complex numbers but with new perspective). And in this paper, I will use the term “function” in all these periods and stages just to avoid difficulties and confusion, and using the term “function” has no influence on the outcome, and I will most of the time stay in the Arabic-Islamic stage.
2 This depends on the field or the branch of science.
3 A figure with a unit circle and all these line segments representing the six functions found in almost every pre-calculus textbook (college level course).
4 In the beginning of this paper, the reader might find a point or two to disagree with. But if not all, most of what I am going to use are somehow regarded as facts among those who worked recently in this field, like M.-Th. Debarnot. A little astronomy will also be employed.
5 With Ārybhata (d. 550) and his works, as Arya-Siddhanta, and also Varahamihira (d. 587), and Bhaskara I (d. c. 680).
6 The idea behind this change from “the chord” to “half the chord” as it is well-known was in the first place the simplicity.
7 See M.-Th. Debarnot, “Trigonometry”, in R. Rashed (ed.), Encyclopedia of the History of Arabic Science, 3 vols. (London and New York, 1996), vol. II, pp. 495–538, p. 495.
8 All sine tables were tablated with radius equal to sixty.
9 This will be explained.
10 He is known in west as Albategnius.
11 I.e. the shadow is on a horizontal plate, and in this case the miqyās is set to be vertical, the other case where the miqyās is horizontal will be discussed later. The vertical or the horizontal cases are chosen to make use of Pythagorean theorem, again a kind of simplification.
12 Onward, I will use the term miqyās.
13 I will replace “half the chord” with “sine”.
14 In a similar situation.
15 The meaning of the number has no importance here. The meaning could depend on the branch of the science e.g. astronomy or astrology, and also the measure of the unit e.g. hair, finger, or foot. Al-Battānī’s field was astronomy and his unit was finger.
16 The miqyās could be outside the circle, and whatever, the calculation will be the same. MN is also the miqyās but in the case where it is horizontal.
17 It is the sine of arc DG, which is half the chord of arc DE, which is double of arc DG, which is the altitude.
18 It is the cosine of arc DG, and AC is equal to EH which is half the chord of arc EF, which is the complement of arc DE to 180.
19 Throughout the paper Sin x = 60· sin x, as well as for the cosine and the chord.
20 Ḥabash could be the first who tablated these ratios in this understanding, but later Abū al-Wafā’ (d. c. 998) tablated them in a much better way as it will be shown. See M.-Th. Debarnot, “The Zīj of Ḥabash al-Ḥāsib: A survey of the MS Istanbul Yeni Cami 784/2”, in D. A. King and G. Saliba (eds.), From Deferent to Equant (New York, 1987), pp. 35–69, p. 35.
21 Three selected problems from many others, this is to show that this technique was widely used.
22 Be aware that at the time of Abū al-Wafā’ all the six trigonometric functions were available as it will be shown. All what I will mention about Abū al-Wafā’ will appear in A. Moussa, Commentary and Edition of Abū al-Wafā’’s Almagest, three volumes, soon to appear successively.
23 Regarding the constants k mentioned in the previous problems (and also others in other similar cases), I personally do not know whether they have meaning in modern astronomy or not, or in any other branch of science.
24 Here the technique was much more advanced than that of al-Battānī.
25 See E. M. Bruins, “Ptolemaic and Islamic trigonometry, the problem of the Qibla”, Journal for the History of Arabic Science Aleppo University (Syria), 9 (1991): 45–68, also, vol. LXXIII (1986–1990).
26 As an example from the late scientists, al-Kāshī (d. 1450) tablated the sine with radius equal to sixty.
27 In his youth, Abū al-Wafā’ studied the works of al-Battānī. This was mentioned in the famous reference al-Fihrist by Ibn al-Nadīm.
28 The talk here also extends to the remaining functions i.e. secant and cosecant.
29 This achievement was a sudden advance and hence it was controversial, so that, in the beginning, some scientists, as Kushyār ibn Labbān and al-Khujandī (d. 1000), did not believe in its truth. Al-Bīrūnī took the responsibility to convince them. This was mentioned by al-Bīrūnī himself in his Keys, see M.-Th. Debarnot, Al-Bīrūnī Kitāb Maqālīd ‘ilm al-hay’a (Damas, 1985).
30 As it is on folio 13 of the Almagest; first section of sixth chapter of Book I.
31 See Debarnot, “Trigonometry”, and see also the Arabic part of Kitāb Maqālīd ‘ilm al-hay’a when al-Bīrūnī discusses Abū al-Wafā’’s spherical tangent rule.
32 In figure 2, there are actually two unequal or two different circles but they coincide, or one circle, where the radius is divided into sixty equal parts regardless its real length i.e. two different radii.
33 He tablated both in the same table with reversing order.
34 Since the miqyās in his table of tangents and cotangents was sixty, which is the same as the base of the sexagesimal system, changing it to be 1, this would be an easy operation, because it would only lead to a displacement of the sexagesimal point one place to the left (or to the right in case of jummal). And Abū al-Wafā’ did mention this. But changing the miqyās to a value other than 1 and 60, this will lead to operations as multiplication and division; Abū al-Wafā’ also explained this.
35 Such as: the plane formulas for addition, subtraction, doubling, and halving, and the spherical law of sines and tangent rule, and also the relations among the trigonometric functions (which he deduced directly from his figure, figure 2).
36 From what already mentioned in this paper, the following should not be a surprise: the unit circle is restricted to the functions tangent, cotangent, secant, and cosecant. In Abū al-Wafā’’s mind still: Sin^{2}x + Cos^{2}x = 60^{2}. The surprise could be this: the newer functions reached the unit circle before the older ones.
37 Al-Kāshī generalized it more by considering the four quadrants, and introduced the term munqih, which corresponds to the “reference angle” in modern times, but of course without taking into consideration the negative values. His table of tangents was also of radius equal to 60 as his table of sines; see J. Hamadanizadah, “Trigonometric tables of al-Kāshī”, Historia Mathematica, 7 (1980): 38–45, p. 40.
38 Not only Abū al-Wafā’ used the value 1, but also others as al-Bīrūnī and Kushyār ibn Labbān.