Читать книгу The Little Book of Mathematical Principles, Theories & Things - Robert Solomon - Страница 33

Trigonometry is concerned with calculating sides of triangles from angles and early development was mainly used in astronomy.

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Trigonometry is taught throughout high schools. It is based on three functions: sine, cosine, and tangent.

The original trigonometric function was the chord function. Start with an isosceles triangle rather than a right-angled one. Let the equal sides each have length one unit. The chord function, crd(Ø), gives the third side of the triangle.

A definition of the chord function.

It is easy to convert between the chord and sine functions, with these formulae, which involve only doubling and halving: crd(Ø) = 2 sin(¹/₂ Ø) sin(Ø) = ¹/₂ crd(2Ø)

No trigonometrical work of Hipparchus survives but he is known to have compiled the first trigonometric table – a table of values of the crd function.

The oldest surviving table is in the Almagest of Ptolemy, a work of astronomy. The table is a feat of numerical complexity: starting with results like crd(60°) = 1 and crd(90°) = √2, and using formulae for crd(A + B) and crd(A – B), the chords of angles are found for every ¹/₂°, to an accuracy of up to 6 decimal places.

The familiar functions of sine, cosine, and tangent, introduced by Indian and Arab mathematicians, are now used but the methods remain the same.