Question

Find all points (if any) of horizontal and vertical tangency to the curve. Use a graphing utility to confirm your results. (Order your answers from smallest to largest x, then from smallest to largest y. If an answer does not exist, enter DNE.) x = cos , y = 2 sin 20 Horizontal tangents (x,y) - Vertical tangents (x,y) - -(( (x,y) -

Answer 1

Here, we have to take derivative of x and y

$dx/d\theta = -sin(\theta)$

$dy/d\theta = 4cos(2\theta)$

So,

$dy/dx = -4cos(2\theta) /sin(\theta)$

When this goes to zero, there will be a horizontal asymptote.

This happens when

$cos(2\theta) = 0$

$\theta) = \pi/4, 3pi/4, 5\pi/4...$

This will correspond to

$(x, y) = (1/\sqrt 2,2), (-1/\sqrt 2,-2),(-1/\sqrt 2,2),(1/\sqrt 2,-2),$The vertical asymtote happens when dy/dt is not equal to zero, and dx/dt = 0

This happens at

$\theta = 0,\pi,2\pi...$

So,