306 EQUATION TO TANGENT. 



280. The result of the last Article may also be obtained 

 thus: 



dr 



sin 6 JQ + r cos 



tan PTx = - ^2 - , (Art. 278), 



COS0-T;; r sin 6 

 do 



PSx = 6 ; therefore 



tt/* 



sin 6 -ja + r cos 6 



do Q 



, tan 9 



adr . a 



cos v -jf. r sin v Ja 



do du . . 



tan SPT= - - = r T- by reduction. 



tan 6 ( sin - fn + r cos # ] 



v ^ 



cos - r sin 



281. To /wcZ the polar equation to the tangent to a curve. 



Let P= r, PSx = 0, be the polar co-ordinates of the point 

 of contact. 



Let SQ = r, QSx 0', be the polar co-ordinates of a point 

 Q in the tangent line. From the triangle SPQ, we have, 

 putting SPQ = (f>, 



r _ sin SQP_ sin (0 - & + <fr) 



r' sin &P sin 



= sin (0 - 0'} cot + cos (0 - &}. 



But tan <& = r T- ; 



ar 



therefore = sin (0-0') + cos (0 - 0') ........... (1). 



