18 AX GEOMETRY 



By means of thest eight relations all the trigonometric 

 functions of any angle may be expressed in terms of any 

 given function. E.g., suppose the sine of an angle is gi\ n, 

 and the tangent of this angle, in terms of the sine, is \\ a n t 1 : 



HJI10 



OOi 



by (4), t) 



and by (6), cos = VI- sin 2 0, 



sin n 



tan 



\ 1 si.rtf 



If the numerical value of sin 6 is given, tins hist formula 

 gives the corresponding numerical value of tau0; e.<j.. it 

 sin = {, then 



tantf 



15. Functions of related angles. Based upon the defini- 

 tions of the trigonometric functions the following relations 

 are readily established. 



If 6 is any plane angle, then* 



(1) sin(-0)= sinfl, cos( 0) = + cos0, 



tan (-0)=- tan 6, csc(- 0)=- cs< 0, 



sec ( - 6) = + sec 0, cot ( - 6) = - cot 6 ; 



sin (TT 0) = T sin 0, cos (TT 0) = - cos 0, 



tan(ir0)=tan0, csc(7r0)= T csc^, 



cot(?r0)= 



(8) s 



tanf^ 0J= T cot 0, csc(^ J= -h sec 0, 



gec^ 0J= T esc 0, cot ( f ^ j = T tan ^ l 



The student should thoroughly familiarize himself with these formulas, 

 and those of Art. 10, as well as with the derivation of each. 



