I don't quite understand how the following works, could anyone please explain?
Let $O_i,i=1,2,3$ be linear operators on a space of functions.
They satisfy the commutation relation $[O_a,O_b]=O_aO_b-O_bO_a=i\epsilon_{abc}O_c$
Why then is $\exp(-i\phi O_2)O_1\exp(i\phi O_2)=O_1\cos\phi-O_3\sin \phi$?
I don't understand how to deal with the exponentials. I thought that maybe it had to do with Euler's formula, but I don't quite know how that works with the operator.
Thanks in advance!
I believe it is related to angular momentum operators in quantum mechanics.