How to calculate rise time of an Op Amp
I'm not giving a full answer because the OP needs to learn.

Saying that the input is a cosine wave is confusing you. The output will have the same rise and fall times if it were a sine wave except now you can start at t = 0, whence the output waveform begins at 0 volts and 0 degrees of phase angle.

If the amplified waveform was unbounded by the +/- 5 volt rails it would reach  peaks of +/- 100 volts. If you superimpose the 5 volt limits you get something very similar to this: -



It's a simple matter of using your brain/equipment to calculate arcsin(0.05) to give you the degrees that have passed to hit a level of 5 volts. 

Can you take it from here?