Radians: Why are units on one side radians/sec and on other side 1/sec?

[Steven G]

Sin u = opp/hyp.
1)Suppose opp and hyp are measured in meters.
2) Suppose hyp is constant.
3) Suppose time is measured in seconds.

Sin u = opp/hyp
cos u * du/dt = (1/hyp)* d/dt(opp).
The units on the lhs is radians/sec.
The units on the rhs is 1/sec.
How can this be?

 

[skeeter]

both sides are [imath]\text{sec}^{-1}[/imath]

radians are unitless ... a ratio of arclength/radius