If `x=3sint-sin3t ,y=3cos t-cos3t ,"f i n d"(dy)/(dx)"a t"t=pi/3dot`

Explanation

We have given
`x=3 sin t-sin 3 t`, and
`y=3 cos t-cos 3 t`
Differentiate `x` with respect to `t`

`frac{d x}{d t}=3 cos t-3 cos 3 t`
Differentiate `y` with respect to `t`

`frac{d y}{d t}=-3 sin t+3 sin 3 t`

`frac{d y}{d x}=frac{frac{d y}{d t}}{frac{d x}{d t}}=frac{-3 sin t+3 sin 3 t}{3 cos t-3 cos 3 t}`

When `t=frac{pi}{3}`

`frac{d y}{d x}=frac{-3 sin (frac{pi}{3})+3 sin 3(frac{pi}{3})}{3 cos (frac{pi}{3})-3 cos 3(frac{pi}{3})}`

`frac{d y}{d x}=frac{-3 times frac{sqrt{3}}{2}+0}{frac{3}{2}-3(-1)}=-frac{1}{sqrt{3}}`