TRIGONOMETRY 83 



T 3C 1 ^C 



16. 7J-T tan = -cot - 2 cot 2x. 



^^^^/ jit /2*^ O O 



& = o 



'V^ k z 2ir \/n / mr mr\ 



17. / cos = i + cos h sin 



J n 2 \ 2 2 / 



. k 2 2ir Vnf mr mr\ 



in = i + cos - sin 



n 2 \ 2 2 



I 



X^ 

 18. /. sin 



4-- 

 n i 



7T 7T 



IQ. x.sm = cot 



n 2n 



2j /^ 2 2n ~*~ 2 _ i IT 



^ tarf ~ k - ..-i + 4 cot* 2* - ^ cot - - 



3.62 



I 



CSC 



Watson (Phil. Mag. 31, p. in, 1916) has obtained an asymptotic expansion 

 for this sum, and has given the following approximation: 



S n = 2^0.7329355992 logio(2w) - 0.1806453871} 



_ 0.087266 0.01035 0.004 0.005 



n n* n b n 7 



Values of S n are tabulated by integers from n = 2 to n = 30, and from n = 30 

 to n = 100 at intervals of 5. 



The expansion of 



27T o 27T 



where -- <p< , 



n n ' 



is also obtained. 



