can radius and degree exist in the same angle I and my teacher had an argument about the result of $\sin(\pi-1)$.
He said that to convert the angle from radius to degree you must replace every $\pi$ with 180 so he said that $\sin(\pi-1)=\sin(180-1)=\sin(179)$. but I think that $\pi$ is a value so $\pi-1$ must be evaluated first which $\simeq$ 2.14159
the convert it to degree by multiplying by $\frac{180}{\pi}$ which $\simeq$ 122.7.
So I'm confused should I treat $\pi$ as Radius and 1 as Degree or they must be the same unit.
 A: They must be the same unit. Your approach is perfectly all right. In case of angles, you must treat $(\pi - 1)$ as radian wholly or degree wholly (although I have not seen $\pi^\circ$ in any book till now).
A: Your rendering appears correct to me.  
Having said that, there may be some additional context which explains why your professor advised you this way.  
Ultimately, we can't provide a definitive answer to questions of the form, "So'n'so says X and I say Y. Who's right?" These questions are best resolved with "So'n'so." 
Plug the equation into a physical or online calculator that allows for the full expression to be seen with it's result. Choose degrees mode and then radians mode. Discuss the results with your professor.  
It could be that what your professor is trying to communicate is the necessity of keeping a single unit in the parentheses.  
Only a discussion with your professor can identify any potential misunderstanding.
A: A good rule of thumb is to forget about unit issues and just define
$$º:=\frac{\pi}{180} .$$
PS: This makes you right, by the way.
