USEFUL FORMULAS. XVll 



0-. Values of functions in series. 

 For A- in arc the following series hold within the limits indicated. 



.t3 , x' x' 



in .V = -V — .- + ^,^ Po^n + 



sm 



6 ' I20 5040 



.r- , .T* x^ , 

 cos .V = I - 2 -r 2^ - ^20 -h • • . , 



— oo < .r < + 00. 



tan .V = .V + 1 x'-{-r5x'-\-^,x'' + 



sec x=i-^hx' + ^x*-\- ^<'.h x' + . . . , 



cot .V = - (i - i Jc^ — 4V ^^'* — ^h x^— . . .), 

 cosec a- = ^ (i + i X- + ^J^ -^'^ + TTi'Wij -'^-'^ + •••). 



arc sin .r = .v + J x' + ^a .r^ + ^^^ ^^7 _|_ . _ ^ 



x^ , :r^ .t' i^ x^ 

 arc tan .r = X 7 ' "r 7" ^ ' ' ' ' 



- I < a- < + I. 

 X = sin .V + ^ sin^ x -\- ^% sin^ x + yf ^ sin" .v + . . . , 



- i- TT < a; < + i TT. 



a: = tan .r — ^ tan^ .v + 5 tan^ .r — -f tan'' a -f . . . , 



- i TT < .r < + i TT. 



log sin X = log X — iJ. (J X- + xio -^'' + 5^- .T^ + • • • )> 

 .r positive and < tt, 

 yu, = modulus of common logarithms. See p. xv. 



log tan x = \ogx-\-,x (' x"' + r^^ x* + ^§§j a:« + ...)» 

 X positive and < V tt. 



h. Conversion of arcs into angles and angles into arcs. 



Denote by x°, x', and x" respectively the angle (in degrees, minutes, or sec- 

 onds) corresponding to the arc x. Then by equality of ratios 



360° _ 360 X 60^ _ 360 X 60 X 60" _ 2jr 



^^ — x' —' x" — x' 



whence 



180° 



x" = X 



X = X 



cc — oc 



