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

Answer:

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

Step-by-step explanation:

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

$r=6cos(\theta )\\\\\frac{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 $\frac{\pi }{2}$

Again differentiating the given curve we get  

$r=6cos(\theta )\\\\ \frac{dr}{d\theta}=-6sin(\theta )$$\therefore -6sin(\theta)=\frac{\pi }{2}\\\\\Rightarrow \theta =sin^{-1}\frac{-\pi }{12}$

$\therefore \theta =-15.17^{o}$

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