Useful deductions from Maclaurins theorem

The following may be considered as a series of examples of the use of the formula obtained in the preceding section. Many of the results now to be established will be employed in our subsequent work. 

In order to expand any function by Maclaurin s theorem, the successive differential coefficients of u are to be computed and x then equated to zero. This fixes the values of the different constants. 

When n is a positive integer, and n = rr,y the infinite series is cut off at a point where n - m = 0. A finite number of terms remains. 

These series are employed for calculating the numerical values of angles between 0 and tt. All the other angles found in trigonometrical tables of sines and cosines, can be then determined by means of the formulas, page 611, 

From page 67, by successive differentiation of this expression, 

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