How can I downscale daily values of relative humidity?
Your approach to calculate an hourly temperature and relative humidity independently could be problematic, as these variables are dependent on one another.  I'd be interested to see a plot of one such day of $t$ and $rh$ and then to calculate the water vapor mixing ratio or dewpoint from your values and see how it varies in response and whether it is realistic looking.

If all you have in mean $rh$ and and a min/max temperature, one approach might be to derive a mean temperature and then from that a mean water vapor mixing ratio.  This mixing ratio can vary through evaporation, condensation, boundary layer mixing, etc, but for a rough approximation you could assume it is constant.  If you assume a constant mixing ratio and a temperature profile (e.g. your sinusoidal assumption) you could calculate the $rh$ profile for the day.  Note that you'll have to assure that your temperature curve does not cool below the dewpoint. You have the mean $rh$ as a crosscheck for this method and can use that to iteratively adjust your method for obtaining the water vapor mixing ratio.  

Your diurnal temperature profile will vary based on latitude, time of year, proximity to large bodies of water, and weather.  If you discount the weather you can get by with your sinusoidal approximation with your minimum temperature at local sunrise and your maximum temperature in the mid to late afternoon.  The amplitude of the sine wave will be dependent on the your location (e.g. coastal locations will have small amplitude changes and inland desert locations will have large changes).  With a constant water vapor mixing ratio this will yield the highest RH at the temperature min at sunrise and the lowest RH in the mid afternoon. 

If you can provide a $t_{max}$, $t_{min}$ and $\overline{rh}$ and a rough idea of where on earth your are looking at, I can run some numbers and provide more detail on this method.