As stated here, the angle
θ in radians is defined by
considering the arc of a circle 




A right angled triangle including θ has
been added to this diagram of a segment of a circle. 



From this, we have 



It should be clear from this diagram that these two ratios are not equal
(s is noticeably longer than y). 

For example, if θ = 60° then 






Here the angle is smaller, maybe about 30°. 

Now we find 



Already quite close... 






Now consider an even smaller angle (diagram enlarged to be able to put the
letters on it!) 



It is now clear that y is nearly the same length as s. 





If this angle is 10° then we now have 



So, even at 10°, the angle is now small enough for
these two ratios to be equal, to three decimal places. 



N.B. 

Remembering that tanθ
is defined as 



we could use similar reasoning to find that for small angles tanθ
is also equal to θ in radians. 



In conclusion, for small angles, (usually considered to mean 3° or less): 


