SINE. COSINE. TANGENT. 31 



51. Differential coefficient o/sin x. 

 Let y = sin x, therefore 



y + Ay = sin (x + h), 



therefore Ay = sin (x + h) sin a; 



h\ . h 



= 2 cos ( x + - ) sin - , by Trigonometry, 



\ 2/ A 



h 



. , . sin - 



therefore -r-^ = cos ( x + - ) * . 

 Aa; \ 2J h 



. h 



sm- 

 Now when h is indefinitely diminished, the limit of 



is unity by Art. 9, therefore 2 



dy 



-/- cos x. 



dx 



52. Differential coefficient of cos x. 

 Let y = cos x, therefore 



y + Ay = cos (x + A), 

 therefore Ay = cos (a; + K) cos># 



/ h\ . h 

 = 2sm la;+ -J sin-, 



c . n k 

 therefore ~^ = si 



therefore -g- sin x. 



dx 



53. Differential coefficient o/tan x. 

 Let y = tan x, therefore 



y + Ay = tan (a; + /*), 



