Chapter 10 - Conics, Parametric Equations, and Polar Coordinates - 10.3 Exercises - Page 711: 23

$$y = \pm \frac{3}{4}x$$

$$\eqalign{ & x = 2\sin 2t,{\text{ }}y = 3\sin t \cr & {\text{The graph crosses itself at the origin }}\left( {0,0} \right) \cr & {\text{At the point }}\left( {0,0} \right) \to t = 0,{\text{ }}t = \pi \cr & {\text{By theorem 10}}{\text{.7}}{\text{. The slope is}} \cr & \frac{{dy}}{{dx}} = \frac{{dy/dt}}{{dx/dt}},{\text{ }}dx/dt \ne 0,{\text{ for this exercise consider }}t = \theta \cr & \frac{{dy}}{{dx}} = \frac{{\frac{d}{{dt}}\left[ {3\sin t} \right]}}{{\frac{d}{{dt}}\left[ {2\sin 2t} \right]}} \cr & \frac{{dy}}{{dx}} = \frac{{3\cos t}}{{4\cos 2t}} \cr & \frac{{dy}}{{dx}} = \frac{{3\cos t}}{{4\cos 2t}} \cr & \cr & {\text{*For }}t = 0 \cr & {\text{Calculate the tangent line at the point }}\left( {0,0} \right) \cr & {\text{Evaluate at }}\frac{{dy}}{{dt}}{\text{ }}t = 0 \cr & {\left. {\frac{{dy}}{{dx}}} \right|_{t = 0}} = \frac{{3\cos t}}{{4\cos 2t}} = {\text{ }}\frac{3}{4} \cr & y - 0 = \frac{3}{4}\left( {x - 0} \right) \cr & y = \frac{3}{4}x \cr & {\text{*For }}t = \pi \cr & {\text{Calculate the tangent line at the point }}\left( {0,0} \right) \cr & {\text{Evaluate at }}\frac{{dy}}{{dt}}{\text{ }}t = \pi \cr & {\left. {\frac{{dy}}{{dx}}} \right|_{t = 0}} = \frac{{3\cos \pi }}{{4\cos 2\pi }} = {\text{ }} - \frac{3}{4} \cr & y - 0 = - \frac{3}{4}\left( {x - 0} \right) \cr & y = - \frac{3}{4}x \cr & {\text{Therefore}} \cr & {\text{At }}t = 0:{\text{ }}y = \frac{3}{4}x \cr & {\text{At }}t = \pi :{\text{ }}y = - \frac{3}{4}x \cr & or \cr & y = \pm \frac{3}{4}x \cr} $$