An essential part of a projected coordinate system is the map projection. It mathematically converts points from a spheroid (or sphere) to a plane. Once the points are converted, if the spheroid information isn't kept, it is not possible to 'unproject' the points back to the spheroid. The spheroid is part of a geographic coordinate system which includes more information like the geodetic datum, prime meridian and angular units. The prime meridian and angular units could be restricted to particular values. Could we then retain only the spheroid information in the PCS definition? Unfortunately, the spheroid information isn't enough to uniquely identify a geographic coordinate system. Most spheroids have been used by many different geographic coordinate systems. For instance, the Clarke 1866 spheroid is used by NAD 1927, Guam 1963, and Old Hawaiian. Including the GCS information also means that the angular units are not restricted to a single definition like decimal degrees. Because of this, any angular map projection parameters use the same angular unit as the associated geographic coordinate system.

If someone tells you that a dataset is UTM zone 10N (which is part of a projected coordinate system), the geographic coordinate system information is missing. Was the original model of the earth WGS84, NAD27, or something else? All of this information is important.