Sine of an angle – by Hindu Maths
L V Nagarajan
Aryabhata (AD 476550) was the first in the line of great mathematicianastronomers known to us from the classical age of Indian mathematics and Indian astronomy. His most famous works are the Āryabhaṭīya and the Aryasiddhanta. The following stanza in Āryabhaṭiya gives a series of 24 numbers and calls them as ArdhaJya differences.
मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व 
घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कलाअर्धज्यास् 
Makhi Bhakhi Phakhi Dhakhi Nnakhi Nyakhi
Ngakhi Hasjha Skaki Kishga Sghaki Kighva
Ghlaki Kigra Hakya Dhaki Kicha Sga
Jhasa Ngava Kla Ptha Pha Cha kalaardhajyas
When decoded into numbers it reads thus:
225, 224, 222, 219, 215, 210;
205, 199, 191, 183, 174, 164
154, 143, 131, 119, 106, 93;
79, 65, 51, 37, 22, 7 : Ardha Jya Differences.
You may imagine a bow. The string tying the two ends of the bow is called Jya (or rope) in Sanskrit. (This sanskrit word Jya, for rope, is the root for Geometry, Geology, Geography, as we know them now). Jyaardha is half of this length.
Please see the figures below. In Figure1, ABC is an arc of a circle. AC is its Jya. AM is ardhaJya or halfJya. It is seen clearly that halfJya, AM is nothing but Sine of angle AOB, multiplied by radius OA.
Considering the Figure2 above, all the vertical steps from bottom to top add up to respective Jyaardhas or halfjyas of the increasing angles. In the above diagram, the angles are in steps of 15 degrees. Hence, Step 1 = R Sin 15; step1+ Step2 = R Sin 30; Step 1 +2 +3 = R Sin 45 and so on up to 90 degrees. These steps are called halfJya differences. The above sanskrit verse gives the stepsizes or halfJya differences for 24 steps of 3.75 degrees each to add up to 90 degrees. Thus it gives a table RSines for 0 to 90 degrees in steps of 3.75 degrees, with R= R sin90 = Sum of all steps 1 to 24 = 3438. These values are found to be highly accurate with the present day values of Sines, as shown in the table and chart given at the end.
The genius of Aryabhata defined lengthMB (refer to Figure1) as UtkramaJya, reverseSine or Versine. Aryabhata proposed accumulation of the above Jyadifferences in the reverse order to get the successive UtkramaJyas, as it is obvious from the above stepdiagram where steps are symmetrical about 45 degrees. Hence KotiJya or (Cos x) was defined by Aryabhata as (1 – UtkramaJya).
Aryabhata has actually devised an algorithm to develop this Sine table. The second section of Āryabhaṭiya, titled Ganitapāda, contains the following stanza indicating a method for the computation of the sine table.
rasi lipthashtamo bhaga: prathamam jyaarda muchyathe
thath dwibhakta labdhon mishritham thath dwitheeyakam
aadyenaivam kramaath pindaan bhaktwa labdhon samyutha:
khandaka: syu: chaturvimsa jyaardha pinda: kramadami.
There are several ambiguities in correctly interpreting the meaning of this verse. For example, the following is a translation of the verse (by Katz) wherein the words in square brackets are insertions of the translator and not translations of texts in the verse.
“When the second halfchord partitioned is less than the first halfchord, which is [approximately equal to] the [corresponding] arc, by a certain amount, the remaining [sinedifferences] are less [than the previous ones] each by that amount of that divided by the first halfchord.”
With my highly limited knowledge of Sanskrit, I can guess a few parts of the above Sutra.
Rasi – 12th part of a full circle (30 degrees).
Ashtomo Bhaga – 8th part (of Rasi), 3.75 degrees
prathamam – First; Jyaardha – Sine; Muchyate – obtained
Dwibhakta – Double or Add to itself;
Labdhon – Profit, Dividend, Quotient;
Mishritam – together; Dwiteeyakam The second;
aadyenaivam – In a similar way
kramath – Successive; Pindan – difference
chaturvimsa – twenty four; Jyaardha Pinda – Sine differences;
Kramadami; recursively
Now let me try to give a contextbased English translation of the above Sutra:
Arc of an eighth of a rasi gives you the first Jya;
That doubled and divided – together gives the second;
Same way successive Jyadifferences, together with quotient,
recurrently give all the twenty four Jyadifferences.
There are 12 Rasis in Earth’s trajectory around the Sun. Hence each Rasi is 30 degrees. A quadrant arc of a circle subtends an angle of 90 degrees at the centre, or 3 rasis of 30 degrees each. 8th part of rasi will be 3.75 degrees. There will be twenty four such 3.75 degree arcsectors in a quadrant, totaling to 90 degrees. An arc (or a bow) of 2 x 3.75 degrees will have a Jya (or a rope). Half of this, is Jyaardha and corresponds to Sine of 3.75 degrees; which will be same as this arc of 3.75 degrees, as per the Sutra. Yes, in modern mathematics, it only means Sin(x) = x, when the angle is as small as 3.75 degrees. Evidently, a basic circle of certain radius must have been considered for this purpose and hence Jya really means Rsin(x). For some reason (to be explained later) the first Jya is taken as 225 :– 4 x 24 x 225 gives, the total circumference of the circle considered as 21,600, with corresponding radius of 3438. The value of Jya of the first angle having been initiated as 225, the Jyas of all the twenty four angles can be found following the above Sutra. Interestingly the Sutra above gives different rules for the first two Jyas and prescribes the recursive rule only from Jya3. (Jya is used synonymously with Jyaardha, which actually represents Sine).
Jya1 = JyaDiff 1 = 225; Jya2 = (225+225) – (225/225) = 449; JyaDiff 2 = 224
JyaDiff 3 (as per the Sutra) = JyaDiff 2 – Jya2/225 = 224 – (449/225) = 222.0044; Jya3 = 671.0044
JyaDiff 4 = JyaDiff 3 – Jya3/225 = 222.0044 – (671.0044/225) = 219.0222; Jya4 = 819.0266
The whole table of 24 Jyadiff’s, as developed by using the above sutra is given in a table below. Even this table gives good values for RSines, though not as accurate as the earlier table as could be seen from the plot.
Now let us check this recursive calculation
Difference, D(n) = Sin (n+1)x – Sin nx
= Sin nx Cos x + Cos nx Sin x – Sin nx
Similarly, D(n 1) = Sin nx – Sin (n 1)x
= Sin nx – Sin nx Cos x + Cos nx Sin x
D(n) – D(n1) = 2 Sin nx Cos x – 2 Sin nx = 2 Sin nx (1 – cos x)
We know, Jya(n) = J(n) = R Sin nx
Hence Jya Difference, DJ(n) = R * D(n)
Similarly, DJ(n 1) = R * D(n1)
Hence, DJ(n) – DJ(n1) = R * [D(n) – D(n1)] = – 2 J(n) (1 cos x)
i.e., DJ(n) – DJ(n1) = J(n)/K,
K being a constant and equals [1/2(1cosx)]
With x = 3.75 degrees, It works out that K = 233.5374.
If we start the recursive process with J(1) = K = 233.5374, we will get a very accurate table of Jyas, but with R = K x 48/Pi = 3568. However with values of Pi and (Cos x) as available in ancient Hindu period, values of 225 and 3438 for K and R were good enough, as can be seen from the plot below. Or was there a reason to choose these values? The whole circle is 360 degrees. Each degree can be divided into 60 minutes. Now the whole circle is 360×60 = 21600 minutes. Hence perhaps, the radius of the circle was taken as 21600/2Pi = 3438. An arc of 3.75 degrees will be 225 units long.
The ancient Hindus however knew the exact values of Jya for angles of 30, 45, 60 and 90 degrees and Aryabhata could have very well used them to apply corrections to the above table as required. Hence the value of R may not be very critical. Aryabhatta’s table of Sines, given earlier, is the corrected and improved version of the table as developed by his own formula, and hence is much closer to actual values, especially at 30, 45, 60, 90 degrees.
The whole table looks as below:
As per the recursive fromula 
Aryabhata’s Final Table 
Actual 

Angle x 
Quotient Jx/225 
Ardha Jya Diff  RSine(n) Jyan  Ardha Jya Diff  RSine(n) Jyan 
R=3438 RSine(n) 

0 
0.00  0  0  0  
1 
3.75  1.0000  225  225  225  225  224.86 
2  7.50  1.9956  224  449  224  449 
448.75 
3 
11.25  2.9822  222.004  671.004  222  671  670.72 
4 
15.00  3.9557  219.022  890.027  219  890  889.82 
5  18.75  4.9115  215.067  1105.093  215  1105 
1105.11 
6 
22.50  5.8455  210.155  1315.248  210  1315  1315.67 
7  26.25  6.7536  204.309  1519.558  205  1520 
1520.59 
8 
30.00  7.6316  197.556  1717.114  199  1719  1719.00 
9  33.75  8.4757  189.924  1907.038  191  1910 
1910.05 
10  37.50  9.2822  181.449  2088.486  183  2093 
2092.92 
11 
41.25  10.0473  172.166  2260.653  174  2267  2266.83 
12 
45.00  10.7679  162.119  2422.772  164  2431  2431.03 
13  48.75  11.4405  151.351  2574.123  154  2585 
2584.83 
14 
52.50  12.0624  139.911  2714.033  143  2728  2727.55 
15  56.25  12.6306  127.848  2841.882  131  2859 
2858.59 
16  60.00  13.1427  115.218  2957.099  119  2978 
2977.40 
17 
63.75  13.5963  102.075  3059.174  106  3084  3083.45 
18 
67.50  13.9896  88.479  3147.653  93  3177 
3176.30 
19  71.25  14.3206  74.489  3222.142  79  3256 
3255.55 
20 
75.00  14.5880  60.168  3282.310  65  3321 
3320.85 
21  78.75  14.7906  45.580  3327.891  51  3372 
3371.94 
22 
82.50  14.9275  30.790  3358.681  37  3409 
3408.59 
23  86.25 
14.9980 
15.862  3374.543  22  3431 
3430.64 
24  90.00 
15.0018 
0.864  3375.407  7  3438 
3438.00 
The above is a complete table of Sines as per ancient Hindu mathematicians. The last column gives the value of Rsine, (i.e. Jya) as calculated using the current accurate values. The closeness of the values can be observed in the following chart.
Dedication:
When I was researching for this blog, I came across the works of late Sri T S Kuppanna Sastri, an expert in Sanskrit and Ancient astronomy. I was naturally feeling proud, since I have met him about 30 years back. He is an uncle of my wife and he is the father of my friend Dr. T K Balasubramanian, a retired scientist of BARC. Sri Kuppanna Sastri was a professor of Sanskrit and Astronomy in several colleges. His renowned major works are two books namely Pañcasiddhāntikā of Varāhamihira and Vedāṅga jyotiṣa of Lagadha. I dedicate this blog to the memory of late Sri T S Kuppanna Sastri.
References:
2. History of Ancient Indian Mathematics, C N Srinivasiengar, The World Press Private Ltd. Calcutta (1967)
LVN/26 Nov 2012
December 4, 2012 at 1:23 pm 
Algorithmic stanzas such as the couple indicated sound like a great way to remember facts. These seem to go beyond the mnemonics such as ACTS or All Silver Tea Cups (to remember the signs of trigonometric functions of angles in the four Cartesian Quadrants) we learn in modern school mathematics. The stanzas seem to hold more information than tricks like “poems” (http://en.wikipedia.org/wiki/Piphilology)…
December 4, 2012 at 1:24 pm 
The reference above is to “piems”…
November 5, 2014 at 5:02 pm 
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