How to have a header image fill the entire space (width and height) and remain responsive?

Hi all. I'm sure this is a simple thing, but despite having some experience with websites and HTML, I cannot for the life of me figure out how to do it with WordPress. I have a header image that I would like to scale to occupy the entirety of the width of the screen while also showing the full height of the image. I can accomplish the "scales to width of window" part easily, but any theme I try cuts off the height of the image and doesn't seem to have any obvious settings to change that. And defining a specific "minimum height" for the header space doesn't work either, because what the minimum height ought to be changes with the width of the window (for instance if someone was using a windowed browser vs fullscreen). I feel like this should be very simple to do, but I'm at a loss. What am I overlooking?