When asked to find a coterminal angle, can you combine radians and degrees?

Question

Upon creating a recent exam, a colleague of mine told me that he would accept the answer $\frac{\pi}{5}$+360$^{\circ}$ as a positive coterminal angle to $\frac{\pi}{5}$. I disagreed and said that you cannot combine radians and degrees without conversion (and also that $\frac{\pi}{5}$+360$^{\circ}$ is not an angle). He says that is just semantics and that 360$^{\circ}$ is in fact, 2${\pi}$. Who is right?

Answer 1

While I don't think the answer is teeeeechnically wrong, it is absolutely improper to combine two uncompatible units as such. You don't tell people that they are $5$ feet and $22$ centimeters tall. I would not award credit -- and I think a reasonable expectation of students that could even arrive at an answer of $\displaystyle \frac{\pi}{5}$ for any question is that they know that $2\pi$ is the same angle as $360º$. You are in the right, and it is important to instill mathematical clarity in students.

Answer 2

I would say you colleague is right (but it's not 100% clear) since it is only a matter of unit. For instance take $\varphi$ and $\theta$ two angles, the sum $\varphi+\theta$ is well defined as an angle, now suppose $\varphi=\frac{\pi}{5}rad$ and $\theta=360^{\circ}$, you can still write $\frac{\pi}{5}+360^{\circ}$ since it a sum of two angles, regardless the way you write them. For example you can add 1 hour and 30 minutes : $1h+30min=1h30min$, same thing.

Answer 3

I completely agree with you. Degrees and Radians are two different measures for angles. They can be converted from one to another but it is wrong to add them up. Let's consider a different example, if you have a binary number 110011 then you can not add a decimal number 50 to that binary number. You just have to convert one to the other system before any mathematical operation.

I would say $\frac{\pi}{5}+360^{\circ}$ is not the coterminal angle for $\frac{\pi}{5}$.