history of mathematics - National STEM Centre

The Arabs
Activity 5.2
Figure 5.3a
O
Figure 5,4
58
3 cm
50°,
Then, in AD 100, Ptolemy used this table to construct his own table of chords.
The trigonometry of chords
1 Ptolemy's chord table contained entries like crd 36°= 0;37,4,55 where the
numbers are in sexagesimal notation.
a Use modern trigonometric methods to find the true value of crd 36°, and
determine whether Ptolemy's value is accurate, correct to three sexagesimal places.
b Ptolemy's tables could be regarded as three-figure sexagesimal tables. What
would the fourth figure have been?
2 How would you use tables of chords to calculate the length x in the right-angled
triangle in Figure 5.3a?
3 Explain how you could use chord tables to find x in Figure 5.3b. (Don't calculate
x. It is a strategy which is required.)
You can see, from question 3 of Activity 5.2, that the trigonometry of chords is
inconvenient when you come to deal with triangles which are not right-angled.
From AD 400 to AD 700, the Indian mathematicians continued the development of
trigonometry, getting close to modern trigonometry. They used figures similar to
Figure 5.4, in which the line AMB was called 'samasta-jya', the bow string, later
abbreviated to 'jya'. By calculating AM, OM and MC, they developed tables for
sin a, cos a and 1 - cos a, called vers a.
'There is a curious story about the derivation of the term 'sine'. The word'!
'jya', when translated into Arabic, became the meaningless word 'jyb' - ;
written, in the Arab custom, with consonants only. When this term was
translated into Latin it was read as 'jaib', meaning bosom! The word
'sinus', meaning a fold in a toga, was used instead to refer to the sine. as
'••S!
The Arab mathematicians were able to develop tables of increasing accuracy as the
following activity shows.
Activity 5.3 Compiling trigonometric tables
In this activity, the only operations for which you should use your calculator are the
usual four arithmetic operations, plus taking a square root, because they were the
only facilities available at this time. These operations were performed by hand and
were time consuming; you have the advantage of using a calculator with a memory.
1 The Arabs calculated the sines and cosines of 30°, 45°, 60° using expressions
familiar to you in terms of square roots. Write down these values.