The question is to determine the horizontal tangent lines to the polar equation: $r=a(1+\cos \theta)$. Setting $dy/d\theta = 0$, I get $\theta= \pi, \pi/3,$ and $5\pi/3.$ I notice that when $\theta =\pi$, then $dx/d\theta $ also is $0$ , thus producing a singular point at which it is inconclusive as to whether there is a horizontal tangent line at $(0,\pi)$ or not. Without graphing (I do know what this simple cardioid looks like when graphed, but often the equations are more complex and no graphing calculators are allowed), how would I check to see if there is a horizontal tangent line there?
Check the case of singular point with polar equation tangent lines
calculus
Best Answer
Evaluate $\displaystyle \lim_{\theta \to \theta_0} \frac{dy/d \theta}{dx/d\theta}$.
If the limit exists, then there is no corner. If it is zero, then there is a horizontal tangent line.