graph polar coordinates $ r=4\sin(3\theta) $

Question

Graph polar coordinates $ r=4\sin(3\theta) $

I was told by my teacher to split the graph into $3$ parts per quadrant and try those angles the problem arises when I plug $\dfrac{5\pi}{6}$ into the equation and I don't know what to do to solve it without a calculator. If someone has a better process to graphing polar equations/curves I would be grateful.

Answer 1

I'm going to answer your question in a more-general context- that way, you'll hopefully be able to sketch any (reasonable) polar curve.

Some useful points:

• Tangents at the pole: these occur when $r=0$ (when the curve is at the origin).

  In this case, tangents at the pole will be the values of $\theta$ satisfying $4\sin(3\theta)=0$ (which I'll leave you to solve).

• Maxima- these are points which are furthest from the origin- maxima occur when $\frac{dr}{d\theta}=0$ (and $\frac{d^2r}{d\theta^2}<0$).

In this case, the maxima are (some) solutions to $12\cos(3\theta)=0$ (which, again, I'll leave you to solve).

• Symmetry- for what value $\phi$ is $r$ periodic? i.e. for what $\phi$ is $r(\theta+\phi) \equiv r(\theta)$?

I'll give you this one for your question- the period is $\frac{2\pi}{\color{green}{3}}$.

That means you can split the graph into $\color{green}{3}$ parts, each of which will be (rotationally) symmetric to the others- one third of the work!

Once you've found and plotted maxima and tangents at the pole, it is now simply a matter of connecting the dots.

The final result is this:

What I've left out are the values of $\theta$ for which $r$ is maximum (in blue), and the values of $\theta$ for which $r$ is minimum (red). Try and do this yourself!