Are you in radians or degrees? Who cares!?

We're to the point of the year where my precalc students are doing more and more things with radian angle measurements instead of just simply in degrees, so I find myself needing to ask more and more often: "Are you in radian mode or degree mode?" when my students give me an answer from their calculators. You see, if students use their calculator to find a trig value for an angle in degrees, but their calculator is set up in radian mode from making the last graph, then they'll end up reporting an answer that's incorrect. But is that answer too high? Or too low?

It got me thinking, is there some sort of an angle that I could give them where they could incorrectly look up the value in radian mode, and yet by sheer coincidence end up with the same value for degree mode?  That is, are there any angles where:
     

To begin, 0 degrees and 0 radians are the same amount, and so all the trig ratios compute correctly regardless of what mode you are in if the angle is 0.  But for angles greater than 0, a degree is significantly smaller than a radian, and so sine and cosine values will quickly vary significantly depending on what mode you are in.

To convert an angle from one form to the other, I typically use the conversion that 180° = π radians, so to convert from radians, multiply by 180/π.  Roughly speaking this conversion is approximatly 57° per radian, or quite roughly, about 60°. Since they are on different sized scales, there is no other angles (besides zero) where an angle in degrees is the same as in radians. Or are there?

Because of the way angles are defined as rotations around the origin, angles greater than 360° end up being equivalent to (technically "coterminal" to) other angles less than 360°.  For example, 400° is, for all practical trig purposes, identical to 40 degrees, which is 40 degrees further than one complete rotation.  The ratio sin(400) is equivalent to sin(40).  This is true for radian angles too.  A complete rotation is equivalent to 2π or approximately 6.28 radians, so an angle of θ = 7 radians is equal to 7-2π or .7168 rad.

Does this brings up the possibility of some overlapping values perhaps?  32 radians is equivalent to .584 radians (after subtracting 5 full rotations of 2π away) and .584 radians is approximately 33.4 degrees.  That suggests to me that there is probably an angle near here where the two would line up exactly!  But how to find it?

Let's let θ be that angle. I want to find the angle such that  or after a conversion:

     

After a little rearranging, I get:
     
and
     .
So 32 degrees is basically coterminal with 32 radians. If I want to guarantee that my students don't get their question wrong on the test because of this issue, I should make sure that I ask them what sin(31.97) is. Or, if I want to make sure they don't get any false positives and get a question correct that they don't really know how to do, I should avoid asking such questions on the tests.

This formula can be generalized by allowing any number of rotations around the circle, and presents a handful of different angles which give the same result whether in degrees or radians, by changing the specific number 5 to any integer n:

which gives the first seven angles of:  0, 6.4, 12.8, 19.2, 25.6, 32, 38.4, and 44.8.  With so many angles where their measure and radians and in degrees overlap, it's a small surprise that in all my years of teaching I've never stumbled across one on accident before!  In my examples, I typically come up with angles in degrees, and every six or so degrees there's a correlation. That suggests one time out of every six where I come up with an angle "at random" I ought to have found one!  I wonder how many times I've done it and a student has gotten answers that "kinda matched" what I had in class but were slightly off?  How many times did I just tell them they rounded wrong?

Now the truth is there are even more opportunities where this could happen when using trig functions because they can overlap even within the same units!  For example, sin(75°) = sin(105°). How many more "angle pairings" could this give?  I didn't take the time to find an equation that allows me to calculate them all, because I imagine one of you readers who's interested could probably do it and type it up in the comments, but I did make a graph showing where the angles occur in the first quadrant:

Graph of:

There are twice as many angles, which I probably could have guessed if I had thought about it a little bit. The angles we found earlier are represented by A, C, E, G, I and K, and it looks like nearly halfway in between each of those is another angle.  This suggests that close to one in every three angles I give as an example for calculations of sine ought to give me results that are similar in either degree or radian mode.