Chain Rule - Using Radians vs. Degrees

A light house is 4m from the shoreline, the light beam rotates counter clockwise at 3revs/min. When the light makes an angle of 60degrees to the shore, calculate how fast the spot of light is travelling on the shore.

I did this by chain rule and go the expressions:
dx/dt = dx/d(theta) * d(theta)/dt
= -4cosec(theta)^2 * (1080 degrees or 6pi)
= 5760 miles/min or 32pi miles/min

The answer expressed with pi is correct only. Where does the theory comes in that makes degrees incorrect, and radians correct - especially as we were using degrees 1st?

Question

Chain Rule - Using Radians vs. Degrees

A light house is 4m from the shoreline, the light beam rotates counter clockwise at 3revs/min. When the light makes an angle of 60degrees to the shore, calculate how fast the spot of light is travelling on the shore.

I did this by chain rule and go the expressions:
dx/dt = dx/d(theta) * d(theta)/dt
= -4cosec(theta)^2 * (1080 degrees or 6pi)
= 5760 miles/min or 32pi miles/min

The answer expressed with pi is correct only. Where does the theory comes in that makes degrees incorrect, and radians correct - especially as we were using degrees 1st?