You have the same outputs on the back of the card though, right?
Looks like the HD 5870's HDMI version supports a maximum resolution of 1920x1200 (So, 1920x1080 for most displays) unlike newer HDMI versions that can support higher resolutions.
Over displayport or DVI though, that card DOES support 1440p resolutions, but if you adapt from the HDMI output on your card to anything else, it's still only going to support 1080p. You need to use either DP from your card to DP on your monitor, or DVI from your card to DVI on your monitor, and preferably if you have to buy a DVI cable you would want to get a dual link DVI-D cable so that it supports the maximum possible resolution and refresh rate for DVI on that card.
Since that TV only has HDMI inputs, but they support up to 4k resolution, then you will probably need to get both a DVI-D dual link cable AND an active DVI-D to HDMI adapter, because I'm not very confident that a passive adapter is going to work in this case. It might, but I'm not positive. One of our other members, Glenwing, could tell you for certain but I can't without looking into a lot further.
So let me ask this, WHAT, exactly, is the GPU to TV connection going to be used FOR? Gaming? Watching movies or other multimedia? Just a second monitor for more real estate?
Because to be honest your best move here would REALLY be to ditch that old graphics card and move up to something newer. If this isn't primarily a gaming system OR if this connection to the TV itself isn't meant for gaming, then even a cheaper graphics card like the GT 730 or GT 1030 would be perfectly capable of supporting that higher resolution, but they aren't strong enough to do any realistic gaming on unless it's very weak games.
Knowing more about the purpose of everything involved, and the full hardware specifications like motherboard, CPU, power supply model, would be really helpful in helping to find the RIGHT solution here.