JOURNAL 



OF THE 



WASHINGTON ACADEMY OF SCIENCES 



Vol. II, JULY 19, 1912 No. 13 



MATHEMATICS. — A table of the circular functions to radian 

 argument. C. E. Van Orstrand, U. S. Geological Survey. 



With the exception of a few tables containing values of the cir- 

 cular sine and cosine to a small number of decimals, there appears 

 to be no satisfactory table to radian argument of these most 

 important and fundamental of mathematical functions. Not 

 only is it frequently necessary to use the higher values in the 

 usual laboratory computations, but they are also needed for the 

 construction of tables of the requisite accuracy in which circular 

 functions are involved in trigonometrical equations or appear in 

 simple relations which are derivatives of other functions. Im- 

 portant examples of such cases are : 



sinh (x =*= iy) = sinh x cos y ± i cosh x sin y 

 cosh {x ± iy) = cosh x cos y ± i sinh x sin y 



gd~ l {x) = j sec x dx. 



Instances of this kind arise from time to time as new appli- 

 cations of mathematical theory are gradually introduced into the 

 various sciences. 



The evaluation of the circular sine and cosine, and many other 

 functions, is readily accomplished by repeated applications of 

 Taylor's series. As a means of forming a basis for the applica- 

 tion of the series in the present instance, 16 values of sin x and 

 cos x between the limits 0.0 and 1.6 were computed roughly to 

 20 places of decimals by direct substitution in the respective 



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