1. The Mataematics Education SECTION B
V ol. VI, No .2, Ju ne 1 972
OT IMP S N$ OF ANCIENT I NDI AN M ATH.No. 2
Indlara Approxlncatlon-cr To Sin€, Coslne And
Versed Slne.
6y R. C. Gupta, Dcpt,ol Mathematics,
Birla Institute Technologlt,
of Mcsra P. O, Mcsra,
RANCHI ( Bihar )
( Re ce ive d l0 A pri l 1972,)
Let the angle A be in degrees. The equivalent of the following rational approximat-
ions to transcendentalfunctions are found in various aocient fndian Sanskrit works
sin.4-4 A , t8t-A) | {.+tti{)0_l(180-/ } (l)
cosl=41 (8100- A2)| (32+00+A2 (2)
Vers l:5A2 | (32400 Az)
+ (3)
Any one o( these can be derived frorn any orher of them. Rules giving these
approximate algebraic formulas lor the trigonometric func;tionsare explicitly found in most
of the important iistronomical and n:athematical works of fndia from the seventh to the
seventeenthcentury A.D.Irr this article we shall give onll few ir:stancesof their ocrurrences,
F r om t he M a h l -B h d s k a ri y a ( w ri tte n about A " D " 6t0 ) of B hi skara I ( nor to be
confusedwith the famous Bhtiskara. II of the twelfth century ) the following text in Sanskrit
may be quotedl.
ssrqt{m€TqtlaqJernt gilrisr: ltistr
a€qgFqeilsgt: olew: <+rrigurGqa: r
sgs{tah Eiqrq lagcteqqo Qdqil?zrt
cSubtract the degreesof the argument from the degreesof half a circle
( i. e., lB0
degrees). Then multiply the remainder by the degreesof the argument and put down the
result at two places. At one place subtract the result from 4C500. B)' oae-fourth of the
remainder ( thus obtained ) divide the result at the other place as multiplied by the
maximum functional value ( i. e. the radius of the circle 2 ...".....'
p /(l B0 -l )
R sinl-
{+o5oo rl (tlo- z1til+
-
Tbat is
Similar rule is found2 in the Brdhma-Sphuta-Siddhanta which was composed in A. D.
628 by Brahmagupta who was a ccntemPorary of Bhaskara I. This rhows that rhe rule had
become well-known in India in the seventh century itself.
The Lilivati ( of BhdsLara ff, circa A. D. ll50 ), whichis rhe most pcpular work of
ancient Indian mathematics, contains a rule3 for finding approximately the length of a
chord in a circle when its arc is given. This rule is essentiallyeguivalent to (r).
In the Buddhi-vilesinl commentary by Ganesha ( 1545 A. D. ) on Lildvatl occuri
thc text4

2. 60 Tbe Mathematics Education
'ger;utrrrartTa: rRqla-oFagw: r
6<re<qtailor eqrqftitu fi'qsqsTn'
tThe square of degrees of the complementary argument subtracted from
the square
( of the degrees ) of the quarter circumference is thc Multiplier and the quarter of that
( former square ) added to that ( latter square ) is the Divisor of the radius. ( The result of
operation is the Sine of the argument ).'
That is, ( 36014P-( 90 - B )2 i4ultiplier, M
-
and ( 360/4 )2 + ( 90 - B ,,14: Divisorr D
then R sinB-R. MID
Which is suitably equivalent to (2) if ( 90-B; is regarded as l.
fn the game context, Ga4esha also gives a ccncise statement for the formula (3) as
follows :
gwtrnrie'a:rlgvr) f.ti,i afnr r
A) qr(r<oFqteeigeme F{rFv;it
rr
.The square of the degrees of the argumeot rnultiplied by 5 is the Multiplier
( while that square ) added to squre of ( degrees of ) half the circle is the Divisor of the
radius. Hcre ( the result of operation ) becomesversed Slne.'
T hat is , R v e rs A:R .5 A' | (1 8 0 2 + # ).
The modern forms of the Indian approximations (l) to (3) may be easily seento be
s inx : l6r ( r -x ) l (5 n 2 } .4 x 2 -4 n x )
c o! 3- ( zrz- 4 r, ) | (r" + z t )
vers r=5.r2/ ( ?Tr+.rz )
Where x is in radians.
These formulas were so popular in Indian that they are found in one form of the
other in almost all original important Indian works of ancient and medieval periods. We
have given just few t1 pical instances. For some more original texts and translations refe-
rence may be made to author's earlier papers where some other mathematical aspectsare
also discussed.
References
t. Maha-Bhdskariya ed. and tr. by K. S. Shukla, Lucknow, 1960; p. 4'5.
2. Brihma.sphuta.Siddhinta ed. by R. S. Sharsra and others. New Delhi, 1966;Vol-III,
p. 999.
3. Lihvati witb Sanskrit and Hindi commentaries by S. R.Jha 4th ed., Varanasi, 1970,
p. 160; and H. T. Colebrooke's English tr. of Lilivatl, reprinted Allahabad,
1967;P . 123.
4 . Lildr at i wit h Bu d d h i -Vi l d s i n i (i n S a nskri t)ed. byV . D . A pte. P art l l , P oona,
1937, p. 213.
5. Gupta, R. C. : Bhiskara I's Approximation to Sine, fndian Journal of Hist. Sci, Vol.2
( Nov em . 1967 ) PP . 1 2 l -1 3 6 .