Question

Find the points on the given curve where the tangent line is horizontal or vertical. (Assume 0 ≤ θ < π. Enter your answers as a comma-separated list of ordered pairs.)r = 6 cos(θ)

Answer 1

point of horizontal tangent is
`(0^(o),6)` and point of vertical tangent is
`(-15.17^(o),5.79)`

Explanation:

For a horizontal tangent it's slope should be zero thus

`r=6cos(\theta )\\\\(dr)/(d\theta)=-6sin(\theta )`
`\therefore -6sin(\theta)=0\\\\ \Rightarrow  \theta =0,\pi`

Thus the ordered pair of
`(\theta ,r)` becomes (0,6) at this point tangent is horizontal

For a vertical tangent it's slope should be
`(\pi )/(2)`

Again differentiating the given curve we get

`r=6cos(\theta )\\\\ (dr)/(d\theta)=-6sin(\theta )`
`\therefore -6sin(\theta)=(\pi )/(2)\\\\\Rightarrow \theta =sin^(-1)(-\pi )/(12)`

`\therefore \theta =-15.17^(o)`

Thus the ordered pair of vertical tangent becomes (
`\theta =-15.17^(o),5.79)`