A little less crude and a little more automated than @Erik's answer:
As with Erik's answer, use a UTM CRS in meters (assuming that's the units for your distances).
Export your shapefile as csv. Under Layer Options->Geometry, select AS_XY. Under Layer_Options->Create_CSVT select yes. Then save.
Open the resulting csv file in your spreadsheet program of choice (eg, Excel). Using trig as Erik suggests (though it was 10th grade in my high school back in the 60s), compute new X and Y values.
If you don't care about the observation point anymore, move the new X,Y cols over the original XY cols and save.
If you do care about preserving the observation point, you will have to edit the csvt file. This is a simple text file that describes the data type for each column and should be fairly obvious what to enter for your new columns. Set the original observation point columns to type Real.
Now open the csv file in Qgis and make sure it's reading the geometry from the columns you want.