Trigonometry Over Arbitrary Fields
Rational trigonometry makes it possible to work in almost any mathematical field (fields of characteristic '2' are excluded for technical reasons) whether finite or infinite. The real numbers are not considered a true algebraic field and rational numbers fulfil their role in relation to a linear continuum. Where the output of a calculation would be the root of a rational number (an algebraic number) it can be added as a discreet element (extending the field) and does not require further evaluation: all results having 'exact' expressions.
Over finite fields, the 'plane' is actually a torus, corresponding to the elements of the cartesian product of ordered pairs, with opposite edges identified. An individual 'point' corresponds to one of these elements and a 'line' (which now 'wraps around' this region) corresponds to an initial point plus all exact multiples of the 'vector' (say '2 over and 1 up') giving the line its direction or slope.
Read more about this topic: Rational Trigonometry
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—Russell Hoban (b. 1925)
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Open unto the fields and to the sky;
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—William Wordsworth (1770–1850)