history of mathematics - National STEM Centre

Figure 5.5
5 Arab mathematics
2 The value of cos 36° was found from Figure 5.5, in which the triangles ABC and
BCD are isosceles.
a By letting BC = 1 unit, and writing down expressions for AB and AC, show that
2cos36°=2cos72° + l.
b Use the relation cos 2 a = 2 cos a - 1 and the result in part a to show that
cos36°= j(l + V5).
c Show further that cos 18°=
d Show how to find the value of sin 1 8
3 The Arabs knew the formulae for sin(a +/3) and cos(a +/3) so they could
calculate the value of sin 12°. How?
They then used the halving technique, shown in questions 2c and 2d, to find sin 6° ,
sin 3° , sin 1 j " and sin j ° and then approximated by interpolation to find sin 1° .
4 Given the values of sin 1 -^ ° and sin -f ° from your calculator, use linear
interpolation to approximate to sin 1°. How accurate is your answer?
A way of improving on the result for sin 1° would be to use a quadratic or a cubic
approximation for the graph of y = sin x° between x = ^ and * = !•£, but a better
method is the one used by al-Kashi in the early 15th century.
Al-Kashi used the formula sin 3 a = 3 sin a - 4 sin 3 a together with the known value
of sin 3° to calculate sinl°. Let x = sinl°. Then 3x -4x 3 = sin 3°. He then re-wrote
this equation in the form x = -U sin 3° + 4.x 3 J and used the xn+1 = f(xn ) method
which you met in the Iteration unit in Book 4 to find x.
5 a Use this method, together with the value of sin 3° from your calculator, to
find sin 1°. Take ^sin3° as your first approximation. How many iterations do you
need before the value of sin 1° does not change in its ninth place?
b Similarly, start from sin 36° to find sin 12°. Why is this method less efficient?
c What happens if you try to find sin 30° from sin 90° using this method?
You can see that, with the arithmetic operations that the Arabs could do:
• compiling a set of trigonometric tables was an enormous undertaking
• choosing an efficient method for calculating a specific result was important.
However, once again it is worth thinking how much easier it was to construct tables
with their 'modern' number system than with the earlier Babylonian system.
Arab recreational mathematics
Problems involving indeterminate equations and arithmetic and geometric series -
both of which appear in various Indian texts whose origins may go back to the
beginning of the Christian era - appear in early Arab mathematics, and form part of
the subject of recreational mathematics of the Arabs. The two most popular
examples of the time were the 'chess-board' and the 'hundred fowls' problems. For
the chess-board problem, al-Khwarizmi, whose name gave mathematics the word
'algorithm', wrote a whole book, and the second problem formed part of the
introduction to Abu Kamil's treatment of indeterminate equations.
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