It is better for my female character to fight with a bow or with a sword?
If I’m being completely honest, I don’t have any particular preference. It’s your character, so, you need to be the judge of what’s an appropriate armament.
Having said that, both is a viable option, if that’s the route you want to go. It’s entirely reasonable for an archer to carry a sidearm, and the sword was a sidearm. Depending on the setting, it may not be the most appropriate weapon, but depending on the setting, a bow may not be a good choice either.
Normally, the sword is a badge of nobility or heroism in fiction. Historically, it was more of the nobility side, though later on it’s use would expand dramatically. This may not apply for your setting, especially if you have a fantasy land loaded with sword wielding adventurers.
The bow’s a similar situation. If your character is a hunter or military archer, then it would make perfect sense for her to carry a bow. While it has less historical validity, it’s also possible your character is a fantasy sniper, or assassin who makes use of a bow.
With that in mind, all of the examples I just listed for archers would also benefit from carrying a sidearm and dagger or knife. Now, if your character is a hunter, with no noble background, and no other reason to carry one, it might make more sense for them to carry a handaxe (and skinning knife.) It’s a natural tool for them to have with them in the wild, as it would help them start fires, and still be effective as a weapon if they found themselves in melee combat. Though, they could carry the axe in addition to a sword. (It depends on how they’re carrying the weapons, and which ones are quickly available in a struggle.)
Ultimately, it’s up to you. She’s your character, not mine. I know what weapons my characters carry, and I know why. But, I don’t know the first thing about your world, I don’t know your character, and I’m in no position to judge how they should arm themselves, just offer suggestions. If she can reasonably carry multiple weapons (which, she should be able to), then having more options is better than less, and there’s no reason one person could not use both a bow and sword.
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