What kind of coax cable are you talking about? Like a TV antenna or cable TV cable?
There is NO SUCH THING as a great looking projector for under $1000. They ALL SUCK HORRIBLY with gray blacks, blurry images (compared to flat-screen TV) and they are WORTHLESS for 4K/UHD and HDR. At under $1000 you won't get more resolution than 1920x1080, but the optics won't be great and TVs don't have lenses. Furthermore, in 10-15 years, projection is going to be DEAD. Expandable LED screens will rule the world because of the larger number of colors, minimal heat generation, long life, clearer images without optics interfering with image quality. For $999 MSRP, the Hisense H9G will DESTROY the image quality of ANY projector even remotely CLOSE in price. This is a nano-dot TV (they call it ULED) with VERY wide color gamut and close to 2000 nits of brightness AFTER calibration. By contrast, a sub-$1000 projector setup with an ACCURATE picture mode and good menu settings will be lucky to produce 140 nits of measured brightness. If you just want a bigger picture, what you need to do is move closer to a 65-inch TV. The big screen MEANS NOTHING. What is important is viewing angle... the amount of your field of vision filled up by the screen. If your big screen setup produces a 40-degree viewing angle, if you move close enough to a UHD TV that you also have a 40-degree field of view, your viewing experience will be IDENTICAL to a large projection screen. But with the 65-inch TV, you get WAY more color and all the luminance capability you need to make UHD/HDR programming (UHD discs, Amazon Prime Video, Netflix, Hulu, and others offer new/original programming in UHD/HDR). DLP projection technology hasn't improved its inherently poor black levels since around 2010 while LCD and LCoS projection technology have made HUGE strides in improving black levels in that time. Calling DLP "black level" black is ridiculous because it isn't even a very dark gray. Put a space movie on the screen and it looks HORRIBLE because the blackness is nowhere near black. Epson makes some of the best inexpensive projectors, but in a $2500 Epson, the replacement lamp costs $300 from Epson, but you can purchase clone lamps for $60-ish. The Epson lamp and the clone look IDENTICAL in every way (I am engineer, trained to notice fine detail differences and design changes) on the outside, but when you put a clone lamp in the Epson projector, there is a beeper inside the projector that goes beep-beep-beep constantly when there is a clone lamp in the projector. Install the Epson replacement... silent projector. No difference in image quality, it's just a matter of Epson screwing with the lamp INTENTIONALLY to force you to buy expensive lamps from them! Why participate in that level of baloney? (I cannot say if Epson does the same with projectors priced under $1000). I have a $25,000 laser-phosphor projector sitting here idle for a year because 65 and 75 inch 2019 and 2020 top-series TVs look so phenominally better than the $25,000 projector that I've lost all interest in projection. And I have a $7000 projection screen that makes images look incredibly better than a cheap screen or a screen painted on a wall. So $32,000 of projection gear is unused for a year (may even be more than a year now) while 8 TVs with MSRPs of $899 to $4000 pass through, every one of them producing images so demonstrably better than the projector (which has FULL 3840x2160 resolution, not pixel shifting like inexpensive projectors you will be looking at) that NOBODY would choose the projected images over the TV images. Note that when projectors say they accept 4K/UHD inputs, that doesn't mean they DISPLAY UHD resolution... the projectors have to down-convert to 1920x1080 (or lower if the projector's native resolution is less than 1920x1080).
There is NO SUCH THING as a great looking projector for under $1000. They ALL SUCK HORRIBLY with gray blacks, blurry images (compared to flat-screen TV) and they are WORTHLESS for 4K/UHD and HDR. At under $1000 you won't get more resolution than 1920x1080, but the optics won't be great and TVs don't have lenses. Furthermore, in 10-15 years, projection is going to be DEAD. Expandable LED screens will rule the world because of the larger number of colors, minimal heat generation, long life, clearer images without optics interfering with image quality. For $999 MSRP, the Hisense H9G will DESTROY the image quality of ANY projector even remotely CLOSE in price. This is a nano-dot TV (they call it ULED) with VERY wide color gamut and close to 2000 nits of brightness AFTER calibration. By contrast, a sub-$1000 projector setup with an ACCURATE picture mode and good menu settings will be lucky to produce 140 nits of measured brightness. If you just want a bigger picture, what you need to do is move closer to a 65-inch TV. The big screen MEANS NOTHING. What is important is viewing angle... the amount of your field of vision filled up by the screen. If your big screen setup produces a 40-degree viewing angle, if you move close enough to a UHD TV that you also have a 40-degree field of view, your viewing experience will be IDENTICAL to a large projection screen. But with the 65-inch TV, you get WAY more color and all the luminance capability you need to make UHD/HDR programming (UHD discs, Amazon Prime Video, Netflix, Hulu, and others offer new/original programming in UHD/HDR). DLP projection technology hasn't improved its inherently poor black levels since around 2010 while LCD and LCoS projection technology have made HUGE strides in improving black levels in that time. Calling DLP "black level" black is ridiculous because it isn't even a very dark gray. Put a space movie on the screen and it looks HORRIBLE because the blackness is nowhere near black. Epson makes some of the best inexpensive projectors, but in a $2500 Epson, the replacement lamp costs $300 from Epson, but you can purchase clone lamps for $60-ish. The Epson lamp and the clone look IDENTICAL in every way (I am engineer, trained to notice fine detail differences and design changes) on the outside, but when you put a clone lamp in the Epson projector, there is a beeper inside the projector that goes beep-beep-beep constantly when there is a clone lamp in the projector. Install the Epson replacement... silent projector. No difference in image quality, it's just a matter of Epson screwing with the lamp INTENTIONALLY to force you to buy expensive lamps from them! Why participate in that level of baloney? (I cannot say if Epson does the same with projectors priced under $1000). I have a $25,000 laser-phosphor projector sitting here idle for a year because 65 and 75 inch 2019 and 2020 top-series TVs look so phenominally better than the $25,000 projector that I've lost all interest in projection. And I have a $7000 projection screen that makes images look incredibly better than a cheap screen or a screen painted on a wall. So $32,000 of projection gear is unused for a year (may even be more than a year now) while 8 TVs with MSRPs of $899 to $4000 pass through, every one of them producing images so demonstrably better than the projector (which has FULL 3840x2160 resolution, not pixel shifting like inexpensive projectors you will be looking at) that NOBODY would choose the projected images over the TV images. Note that when projectors say they accept 4K/UHD inputs, that doesn't mean they DISPLAY UHD resolution... the projectors have to down-convert to 1920x1080 (or lower if the projector's native resolution is less than 1920x1080).