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trigonometric tables can be be used for this purpose. but to calculate the values without any external help, one can use the traditional method of derivation using different geometrical figures. for example the value of sine 600 can be calculated using an equilateral triangle. the ratio of altitude to half the length of side is (sq.root of 3)/2. but that is also sine600. but that is a tedious method of calculation. it is much easier to remember the rational or irrational values of standard angles like 300 450 600 750 900. the values of few other angles can be calculated from these using trigonometric identities and other formulae.

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