In "Course: Oblivion", the duplicate of Chakotay indicates that Voyager has not been traveling in a straight line. The most logical conclusion would be that they were traveling along one of the galactic arms in a kind of S pattern toward, and eventually away from, the galactic core. Alternatively, they could be traveling down an arm until they were close enough to Federation space to "make a run" across one of the "null" regions between two arms/spurs.
Either would make sense since they would need to stop for supplies, fuel, repairs, maintenance etc. and the best places to do that would be in areas where there were stars, planets and, hopefully, advanced civilizations.
In "The Gift", Kes throws Voyager "about 10 years" (9,500 LY) further along on their journey and safely past Borg space. In "Night", they find a wormhole that takes them to the other side of a 2,500 LY "void" in space (don't get me started) which should equate to at least another 2 years. In "Timeless", they use the slipstream drive and take another 10 years off their trip (somewhere between 9,000 and 10,000 LY based on previous statements). Finally, in "Dark Frontier", they use a Borg transwarp coil to go another 20,000 LY, which Janeway oddly estimates to be 15 years (odd since up to now 1,000 LY = roughly 1 year).
Based on statements made on screen they should be about 42,000 LY further along by the middle of season 5 not including their normal progress which should, in itself, account for between 3,000 and 4,000 LY (since they like to stop and poke at stuff). That accounts for a great deal more than half of their trip by the middle of the 5th season.
Assumptions:
- According to Memory Alpha, the Milky Way is 100,000 LY wide.
- The Delta Quadrant borders the Beta Quadrant and the Gamma Quadrant but Voyager set a course for Earth, not the Idran system (Gamma Quadrant side of the Bajoran wormhole) which implied that would be a shorter trip.
- The Beta Quadrant was mentioned a handful of times in the series, the Gamma Quadrant was (almost) never mentioned which implied they were going through the Beta Quadrant.
- The would have to go from one border of the Beta Quadrant to the other.
- Earth/The Sol System lies on the boundary between the Beta and Alpha Quadrant
- The Caretaker array was closer to the Beta/Delta Quadrant boundary than the Delta/Gamma Quadrant boundary.
All of that said, shouldn't Voyager have crossed the boundary into the Beta Quadrant while they were using their stolen transwarp coil in "Dark Frontier" at the latest?
Answer
The short answer is that they shouldn't have still been in the Delta Quadrant, but that would have dramatically conflicted with the average viewer's understanding of their situation.
In an interview with Ex-Astris-Scientia, Rick Sternbach (Star Trek Senior Illustrator and Technical Advisor) dealt with this specific question in a Q+A;
I can only guess that the writers simply wanted to keep only the Delta Quadrant in people's minds, without having to deal with anything new. They didn't deal with what I call "situational tech" like Voyager's location/trajectory terribly well, even with adequate explanations and diagrams.
Most trek nerds agree that since Janeway confirmed (in VOY: Renaissance Man) that the R'Kaal control "thousands of parsecs from here to the edge of the Beta Quadrant" and that since Voyager effectively jumped them (in VOY: Dark Frontier), that Voyager actually entered the Beta Quadrant at the end of season 5...
Janeway (V/O) : Captain's Log, Stardate 52619.2 : We got another 20,000 light-years out of the transwarp coil before it gave out.
... which places the entire 6th and 7th season in the Beta Quadrant.
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