Qadi Zada computed sin 1 [degrees] to an accuracy of 10-12 (if expressed in decimals), as did al-Kashi.
J. J O’Connor and E.F Robertson: Arabic Mathematics, a forgotten brilliance;
sine of one degree:
First, using the formula
sin(a/2)=sqrt((1-cosa)/2)
and the already known value of the cosine of 36 degrees, the value of sin18 and therefore cos18 can be calculated. Repeating the same procedure, sin9 can be found. Then, using the formula
sin(a-b)=sin(a) cos(b)-sin(b) cos(a)
for a=45, b=18, the value of sin27 is obtained. Using the same formula for a=30, b=27, we arrive at the value for sin3.
At this point, the formula
sin(3)=3sin(1)-4(sin(1))^3
can yield the value for x=sin1, provided we can solve the equation
4x^3-3x+q=0, where q=sin3.
The methods of solving cubic equations however, such as the one considered here, were not available to 15th century mathematicians. The Samarkand scientists then proceeded by rewriting the equation in the form x=(q+4x^3)/3 and establishing a sequence with first term x1=q/3 and n-th term xn+1=(q+4(xn)^3)/3 which indeed converges to the solution (note that the first term of the algorithm was selected as a linear approximation of sin1). Thus the final step of the calculation involves the application of an algorithm and the approximation of the solution, a task referring to the modern Numerical Analysis.
The above described method, although ingenious, is nevertheless painstaking and laborious as it involves the handling of numbers with several decimal digits in order to achieve the desired accuracy. The amount of labor invested by the Samarkand group in compiling only the table of sines can be appreciated if we consider that their table had an increment of only 1 minute (1/60 of the degree) and therefore it consisted of 5401 entries, with an accuracy of 5 sexagesimal (base 60) digits (equivalent to 9 decimal places, which is the accuracy provided today by most pocket calculators).
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