Generate a clock signal at (say) 16x the 10kHz, and use a DAC to stairstep approximate the sine. At 16x, you will only have harmonics present that are > 160kHz, and so very easy to filter.

At 16x, in fact,  you only need 4 or 5 DAC values, assuming you do the 180°-360° portion by inverting the first half. You can build the DAC with just a string of a few resistors and tapping off (taps at sin(15°), sin(30), sin(45) etc.). Send the output of the DAC simultaneously through 2 opamps -- one a follower, and the other a gain of -1. Using logic, alternativerly select the +1 and the -1 path.