Question

Find the slope of the tangent line to the polar curve: r = = 2 cos 6, at 0 = 1 Find the points on r = 3 cos where the tangent line is horizontal or vertical.

Answer 1

An' It may help to sumember that Ma a cost is Use conversion equations a circle . x=4 Coro I sino as a 2 y = way to get dy da slope of tangent line = dy da here M=2680 - a = 4 cos = (2080) GODO = 260820 da 4 Cogol-Sino) -2.2 loso. Sino =-2 Sinzo d Ca si y = = 1 Sino = 120000) Sino = Sin 20 2 Sinecos X = Sin 2x] dy do s 2 Cosad 2 Cos 20 Cosao = dy dy = do da da do Sin 20 - 2 sinzo

dy da at 0= 1/2 Coga! Cooles) -Sin 2 (1) -S(9) -t/ VE 2 dy = da at O=T *+ 8- 13 3 Let M = a coso is a circle x=l cos o y = Elsino > here, 4236000 X- =l caso = 36080) Coso = 360820 da do 3.26080(- Sino) =-3.2 Sino.coB 0 = -3 Sin 20 ( 2 Sin 2 cop2 = S/n 22 y = M Sino = (3000) SinO 2 3.1. Sin 2 O dy 3.1.2 60020 - 3 C0020 do q 360020 Cop 20 11 dy da dy do da do -3 Singo - Sinao

Horizontal tangents ocowe when numeratose.log is 0. We know that M23 logo is a circle dy da there are only two hosizontal (and 2 vertical) to find at more angles results tangents trying ruperting in the same points O= Cop 20 Зл 20= 0 2 2 Q = TT 39 4 4 the e values at these at these angles 3 3.1 그 4= 3 COBI 4 pollo 13 3 4= 3.60 39 = 3.62) where the оссия denominatas A vertical tangents of dy would be o da

0=-Sinzo 2020, IT 0-0 log the de values at these angles we know 34= 3680 = 3()=3 vé 2 4 = 300() - 310) 20 points where tangent line is Horizontal : ), (2) veutical (3,0), 10,7)