[Math] Find the points on the given curve where the tangent line is horizontal or vertical

Question

$r=e^θ $ $($ Assume $0 ≤ θ ≤ 2π.)$

Apparently I keep getting this answer wrong. I dont know if i need to use $n $ in the answer or not…

Answer 1

In Cartesian form $y=e^\theta\sin\theta$ and $x=e^\theta\cos\theta$

Now, $$\frac{dy}{dx}=\frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}}=\frac{\sin\theta +\cos\theta}{\cos\theta - \sin\theta }$$

Tangent is horizontal when numerator is $0$ which gives $\theta=7\pi/4,3\pi/4$

and Tangent is vertical when denominator is $0$ which gives $\theta=\pi/4, 5\pi/4$