I was trying to sketch the polar curve $r=2\sec(\theta)+1$ in the $x-y$ plane. A key aspect in drawing this graph is identifying that the graph has an asymptote at $x=2$. Initially, I was not able to identify this. It took me several attempts to identify this and draw the graph. If I had figured this out earlier it would have been much easier to draw the graph.
Additionally, I find sketching some polar curves like $r=2\sec(\theta)+1$ quite difficult as I am unable to predict the shape without calculating several cumbersome $\sec \theta$ values.
Please could you help me out with 2 things:-
- How to identify asymptotes of polar curves
- Easy and efficient methods to sketch polar curves.
Please note: I am in high school, so please limit your answers to elementary methods.
The asymptotes to a polar curve usually occurs when there's an undefined value. In your equation, from $0 \leq \theta \leq 2 \pi$, $\sec \theta$ is undefined at $\pi/2$ and $3\pi /2$ as $\cos \theta$ is $0$ at both those values (as $\sec \theta = 1/\cos \theta$).
As for easy and efficient ways to sketch polar curves, there are some things to remember...
Symmetry. If you can substitute $-\theta$ or $-r$ in your polar equation and you get the original equation back, it will be symmetric to either the polar line $\theta = \pi/2$ or the pole ($r=0$).
Zeroes and maxima/minima: Setting $r = 0$ and solving will give you critical points for your equation, and thus give you maximum and minimum values for $\theta$. For your equation, setting $r = 0$ will give you $\sec \theta = -1/2$ or $\cos \theta = -2$, but since $\cos \theta$ falls between $-1$ and $1$, there is no solution, and hence there are no maxima or minima for your equation.
There are also polar curves to investigate - polar roses, limaçons, cardioids, leminscates, and so on. Your trigonometry textbook will explain ways of graphing them, or you can see this article for more information.