10 DIFFERENTIAL AND INTEGRAL CALCULUS. 



the rule that sin" 1 ^), cos" 1 (a?), tan" 1 (a 1 ), shall always be 

 interpreted as acute angles, positive or negative ; i. e. as 

 lying between JTT and |TT. This rule will avoid many 

 perplexities and risk of errors. 



therefore x = a siny, x -f Ax = a sin (y + A?/), 



or Aa? = a {sin (y -f A?/) sin y\ = 2a sin -~ cos (y +- r ] 



2 \ 2 / 







Ay 2 



1 = a i . cos 



Aa: A?/ 



therefore l=a~ cosy, -^ = - - = -r/-^ -=r . 

 aa; 7 ax a cos?/ \J(ax] 



The angle ?/ being by our rule between -f ^TT and ^TT, 

 its cosine is positive. If that rule be not followed we must 

 write 



^ = _J _ 



dx V(a'-^)' 



(10) y = cos' 1 - , x a cos#, whence as in the last 



dy . dy 1 



1 = a ~ sm y. -~ = -r. 

 dx yi dx 



This might also have been deduced from the identity 



(11) y^ta 



dy dy a 



(12) 



