Question

find the horizontal and vertical tangents to

x

=

cos

(

3

t

)

and

y

=

2

sin

(

t

)

?

Answer 1

$$\dfrac{dy}{dx}=\dfrac{\dfrac{dy}{dt}}{\dfrac{dx}{dt}}=-\dfrac{2\cos(t)}{3\sin(3t)}$$

Horizontal tangents occur when the derivative equals 0.

$$-\dfrac{2\cos(t)}{3\sin(3t)}=0=>t=\dfrac{\pi}{2}+\pi n, n\in \Z$$

$x=0, y=\pm2$

Horizontal tangents: $y=-2, y=2.$

Vertical tangents occur when the derivative is undefined.

$$3\sin(3t)=0=>t=\dfrac{\pi n}{3}, n\in \Z$$

$x=\pm1, y=-\dfrac{\pi}{3}, 0, \dfrac{\pi }{3}$

Vertical tangents: $x=-1, x=1.$