I don’t remember this particular aspect when I was reading about this, but I struggle to think of cases in which our crafts travel for any significant amount of time perpendicular to our line of sight to them. By definitition this cannot happen for too long, because either the craft flies straight and we revolve or the craft has arrived and it revolves too. In the frist case the angle will not stay at 90 for too long, in the second case we see it straight on at regular intervals.
For the brief periods the ship might be perpendicular we can use intertial navigation.
The other thing is that I suspect geometry can be used pretty close to 90 deg. Cos(80) is still significantly large at 0.2.
Given the incredible precision and accuracy of atomic clocks, one might be able to use the phase of the microwave transmissions from a spacecraft as the signals arrive at two earth stations to calculate position.
At roughly one nanosecond per foot of travel, the two stations might be able to resolve the difference in distance to the spacecraft to within a foot using a 1 GHz signal.
The angular precision of the position of the spacecraft in the sky might then be accurate to one foot in about 8000 miles. This would pin down the position of the craft to perhaps 1000 feet at a distance of 8,000,000 miles, or, extrapolated to a distance of 800 million miles, the position could be determined to within 100,000 feet, or about 20 miles.
Perhaps such a system could be ten times more precise than this, yielding an accuracy of 2 miles.
I can see where this would suffice for just about any purpose. Now I need to find out if this is really what happens.