Question

You are given the parametric equations x=2cos(θ),y=sin(2θ).x=2cos⁡(θ),y=sin⁡(2θ). (a) list all of the points (x,y)(x,y) where the tangent line is horizontal. in entering your answer, list the points starting with the smallest value of xx. if two or more points share the same value of xx, list those points starting with the smallest value of yy. if any blanks are unused, type an upper-case "n" in them.

Answer 1

The tangent line is horizontal whenever
`(\mathrm dy)/(\mathrm d\theta)=0`.

`(\mathrm dy)/(\mathrm d\theta)=2\cos2\theta=0\implies2\theta=\frac{(2n+1)\pi}2\implies\theta=\frac{(2n+1)\pi}4`

where
`n` is any integer.

I'm guessing you're only interested in one complete loop of the lemniscate. In that case, we're restricted to
`0\le\theta\le2\pi`. Then we get 4 points of interest for
`n=0,1,2,3`:

`\theta=\frac\pi4,\frac{3\pi}4,\frac{5\pi}4,\frac{7\pi}4`