I downloaded a Hex editor but I can't find the line that needs to be modified. Is the Gog exe different? Is there a way to play the game in widescreen?

For RCT1, the answer is a definitive no. The game's engine is limited to small 4:3 resolutions and the hex editor may only help you raise the resolution to a certain point, but not make it widescreen.

Well, that's a bummer. Thanks anyway.
I'll check this OpenRCT, also I guess it's time to give RCT3 a shot, heh.

Sorry to contradict you, but there is a way ;) (I'm playing the GOG's french version of RCT1 in 1280x720)
You can look mainly on wsgf : here.
And here are some additional notes from a thread located here in GOG : https://www.gog.com/forum/rollercoaster_tycoon_series/hack_run_rct_full_screen_at_any_resolution_up_to_1280x1024/post37 and https://www.gog.com/forum/rollercoaster_tycoon_series/hack_run_rct_full_screen_at_any_resolution_up_to_1280x1024/post48.

Hope this will help you !

I've seen this method before but I could not find any of the specified strings on the executable using the Be.HexEditor. =/
I wish there was an easier method, like a patch or source port.

You're not looking for strings, you're looking for hexadecimal digit pairs that should be on the left side of your screen. I'm using EditPad Pro for the (little) hex-editing I do. Shows you both at the same time (dunno about Be.HexEditor).

Plokite_Wolf described your problem well, and I was having the same at first (I was stuck during hours too, not understanding what I was doing wrong) : in Be.HexEditor, click on "Find", here, you select "Hex" (and not "Text"), and now you must put the numbers you want to look at, by hand, one after the other (if you're looking for 58 02 00 00 68 20 03 00 00, type yourself 580200006820030000 : the spaces between numbers will be automatically added because you're in hex mode)(a copy/past from the value doesn't work because it convert/calculate the pasted value in Hexadecimal, so it won't be the same number)
Now that you have found the "chain of numbers" inside the file, you modify the numbers you need... You save, and do it again with the second chain of numbers, save, and it should be ok.

But I strongly recommend using wxHexEditor, I found it a lot easier/convenient to use, because there is a "Find & Replace" tool which is pretty convenient (in the edit menu). Go on Edit->Replace, select Hex Mode, in "Search" field, you can directly past the value you want to replace/search(or type it by hand) but remove the spaces between the numbers ! In "Replace" field, same : put the value you want. Click on "Replace all". It should say "1 string replaced" (or something like that). Do the same with the second string you have to replace, save your file.

Once you understood how to manipulate the hex editor, it's in fact easy ! (it's always easy to do something when you know how to do it ;))
Hope this will work now ;)

Edit : I attached a screesnshot of the "Replace" window of wxHexEditor to show how it works exactly (the exemple is for having a resolution of 1280x720).
Value your want to replace :
580200006820030000
By the value :
D00200006800050000
You now have only to click on "Replace All". You should have confirmation that 1 string has been replaced.
After that, you'll have to look for the value 20030000C745F458020000 , and replace it by the value 00050000C745F4D0020000 (if I'm not mistaking in one number). Click on "Replace All", it should confim that 1 string has been modified. Save your file, and it should be ok to play in a 1280x720 resolution ! (just remember to put the in-game resolution to 800x600)

Edit2 : And if you play in another language than english, don't change the name of the modified .exe file : for me, it was making the game to launch in a window automatically if it wasn't the original "RCT.EXE" filename.

Thanks for the detailed explanation but I tried both searching as string and as hex, but no result was found. I downloaded the other editor you mentioned and tried again but nothing. Then I went back to the game and changed the resolution back to 640x480 thinking that the string/hex value should be different for different resolutions, then I tried again but again I found nothing. Then I removed pairs from the beginning and end to narrow the search but again... nothing.
I'm using the english version if that changes anything (probably not). At that point it's too much work just to play widescreen, at least the game works albeit "stretched".
Maybe GOG could released a lancher or a modified exe, but probably won't. =P

Searching as string does'nt work, that's normal.
I don't understand why you don't find the values at all... Just to be sure : you're editing RCT.EXE in the GOG's installation folder, from a "fresh and clean" installation of latest GOG verison ? (seems dumb, but well, sometimes we do evident mistakes).
And with wxHexEditor, go in Edit->Find, select "search as Hex" mode, and past this :
580200006820030000
(beware to remove eventual space which could have been paste at the end/after the huge number) click on "Find all". Nothing ?
If no, can you look for this value (it's the value for applying the widescreen-window hack) :
0500000F8E07000000C745FC000500

Same for me. The GOG RCT.EXE is 791KB and does not contain those hex values. Only the 4+MB version from retail disc seem to have that.

That's strange, mine is 4,79 Mo.
I'm thinking at something : have you installed it with galaxy ?

Hello! I know this is years later, but I ran into this same exact issue with my monitor being 2560x1440. I was able to fix this by changing the monitor's resolution to 1920x1080 in the OS. RCT ran just find in the maximized window mode at 1920x1080.