If `x=3sint-sin3t` , `y=3cost-cos3t` , find `(dy)/(dx)` at `t=pi/3` .

Answer

N/a

Explanation

`:' x=3sint-sin3t` and `y = 3cost-3cos3t`
`:.(dx)/(dt) = 3.(d)/(dt) sint-(d)/(dt)3t=3cost-3cos3t"......"(i)`
and `(dy)/(dt) = 3. (d)/(dt) cos t-(d)/(dt) cos3t`
`=-3sint+sin3t.(d)/(dt) 3t`
`(dy)/(dt) = 3 sin 3t - 3t sint"....."(ii)`
`:. (dy)/(dx) = (dy//dt)/(dx//dt) = (3(sin3t-sint))/(3(cost-cos3t))`
Now, ` ( dy/dx)_(t=pi//3) = (sin'(3pi)/(3)-sin'(pi)/(3))/((cos'(pi)/(3)-cos3'(pi)/(3)))= (0-sqrt(3)//2)/(1/2-(-1))`
`= (-sqrt(3)//2)/(3//2) = (-sqrt(3))/(3) = (-1)/(sqrt(3))`