I posted last summer on a surprising discovery of a polar function that appeared to be a horizontal translation of another polar function. Translations happen all the time, but not really in polar coordinates. The polar coordinate system just isn’t constructed in a way that makes translations appear in any clear way.

That’s why I was so surprised when I first saw a graph of .

It looks just like a 0.5 left translation of .

But that’s not supposed to happen so cleanly in polar coordinates. AND, the equation forms don’t suggest at all that a translation is happening. So is it real or is it a graphical illusion?

I proved in my earlier post that the effect was real. In my approach, I dealt with the different periods of the two equations and converted into parametric equations to establish the proof. Because I was working in parametrics, I had to solve two different identities to establish the individual equalities of the parametric version of the Cartesian x- and y-coordinates.

As a challenge to my precalculus students this year, I pitched the problem to see what they could discover. What follows is a solution from about a month ago by one of my juniors, S. I paraphrase her solution, but the basic gist is that S managed her proof while avoiding the differing periods and parametric equations I had employed, and she did so by leveraging the power of CAS. The result was that S’s solution was briefer and far more elegant than mine, in my opinion.

**S’s Proof:**

Multiply both sides of by *r* and translate to Cartesian.

At this point, S employed some CAS power.

[Full disclosure: That final CAS step is actually mine, but it dovetails so nicely with S’s brilliant approach. I am always delightfully surprised when my students return using a tool (technological or mental) I have been promoting but hadn’t seen to apply in a particular situation.]

S had used her CAS to accomplish the translation in a more convenient coordinate system before moving the equation back into polar.

Clearly, , so

.

In an attachment (included below), S proved an identity she had never seen, , which she now applied to her CAS result.

So,

Therefore, is the image of after translating unit left. *QED*

Simple. Beautiful.

Obviously, this could have been accomplished using lots of by-hand manipulations. But, in my opinion, that would have been a horrible, potentially error-prone waste of time for a problem that wasn’t concerned at all about whether one knew some Algebra I arithmetic skills. Great job, S!

**S’s proof of her identity, ** :

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