Question

The parametric curve described by the equations x=\cos(t),\;\;y=\sin(t)\cos(t) has two tangent lines at (0, 0). find the equations of these tangent lines. list them in order of increasing slope.

Answer 1

Hello,

$x=cos(t)==\ \textgreater \ \dfrac{dx}{dt} =-sin(t)\\ y= \dfrac{sin(2t)}{2}==\ \textgreater \ \dfrac{dy}{dt} =cos(2t)\\ \dfrac{dy}{dx} = \dfrac{ \frac{dy}{dt}}{ \frac{dx}{dt} } =- \frac{cos(2t)}{sin(t)} \\ For\ x=0, \ t=cos(0)= \frac{ \pi }{2} +k \pi \\ ( \dfrac{dy}{dx} )_{x=0} =\pm\ 1\\ tangents \ are\ \{ y=-x,y=x \}$