All you asked for was "transcendental functionS" (Cap "S" mine).

Since the LOCI-1 also did mutiplication and division by logs, I submit that it did, in fact, do transcendental functionS (...just not the ones we might usually think of...).

I did some programming on the Wang 500 series in 1971 specifically for the trancendential functions including trig and log. I seem to recall that most of the functions iterated only to 6 places and we needed at least 9. The little hp35 had the functions with enough precision, but was not programmable.

Hmm... I've got a Wang 520 sitting here and it seems to be at least as accurate as the HP-35 for log and trig functions. It computes everything to 10 digits and compares quite well with the 32SII. (However, I have to admit that the inverse trig functions on this particular machine don't work -- there's a broken ROM wire in there somewhere.) It's my favorite (non-HP) calculator!

Problem was not in the number of digits displayed but the actual accuracy in generating the trans functions. I just don't remember how the Wang stacked up.
Since the question of oldest desktop does not preclude analog computing or accuracy, then there are several slide rule candidates for the first trig/log calculator such as K&E's TrigLogLog which I still own [gr&d].