We present a well-designed sample of more than 1000 type 1 quasars at 3.5<z<5 and derive UV quasar luminosity functions (QLFs) in this redshift range. These quasars were selected using the Sloan Digital Sky Survey (SDSS) imaging data in the Stripe 82 and overlap regions with repeat imaging observations that are about 1mag fainter than the SDSS single-epoch data. The follow-up spectroscopic observations were conducted by the SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS) as one of the BOSS ancillary programs. Reaching i~21.5mag, our sample bridges previous samples from brighter and deeper surveys. We use a 1/V_a_ method to derive binned QLFs at 3.6<z<4.0, 4.0<z<4.5, and 4.5<z<4.9 and then use a double power-law model to parameterize the QLFs. We also combine our data with literature QLFs to better constrain the QLFs across a much wider luminosity baseline. The faint- and bright-end slopes of the QLFs in this redshift range are around -1.7 and -3.7, respectively, with uncertainties from 0.2 to 0.3 to >0.5. The evolution of the QLFs from z~5 to 3.5 can be described by a pure density evolution model ({propto}10^kz^) with a parameter k similar to that at 5<z<7, suggesting a nearly uniform evolution of the quasar density at z=3.5-7.