Question

Given x = cost y=sin2t, how do you find the dy/dx terms parameter t and find the values parameter t points dy/dx = 0?

Answer 1

`(dy)/(dx)=0` at `t=(2m+1)pi/4`

Explanation:

In parametric equations `x=x(t)` and `y=y(t)`, `(dy)/(dx)=((dy)/(dt))/((dx)/(dt))`

As `y=sin2t`, `(dy)/(dt)=cos2txx2=2cos2t`

and as `x=cost`, `(dx)/(dt)=-sint`

Hence `(dy)/(dx)=(-2cos2t)/sint`

As `sint!=0`, when `t=npi`

`(dy)/(dx)=0`, when `cos2t=0` but `t!=npi` i.e. `2t=(2m+1)pi/2`

or `t=(2m+1)pi/4`, where `m` is an integer

But note that `x` and `y` both are sinusoidal functions and hence their domain is limited to `[-1,1]` and hence as `x=cost`, `(dy)/(dx)=0` at `x=+-1/sqrt2`

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