Q: imac to hdtv connection problem

Hopefully the correct HDMI port is selected on the TV...unplug and replug the adapter from the iMac. You may need to reboot your computer. If it doesn't work, you could just have a bad adapter.

For this application you can use a mini-displayport adapter or cable to plug into a thunderbolt port.

Set the TV to the input you connected the HDMI cable to.

On your computer, if you open the Displays preferences and do not see an Arrangement tab (or there is an Arrangement tab that shows only a single monitor) then there is something physically wrong with your connection.  Try clicking Detect Displays just to see if the computer can now sense the presence of the connection. Check your cables and make sure the mini-displayport plug is pushed firmly in so that essentially all the metal part disappears inside the socket.  Mini-displayport plugs can be a little tricky in that you need to make sure the plug is fully seated in their socket.

If you do have an Arrangement tab, and it shows two screens, then click Gather Windows to bring the TV's Displays preferences on to your main screen if the TV is not yet showing an image.  Set the proper resolution (probably 720p, 1080i, or 1080p).  Set the refresh rate (probably 60Hz) if that preference is shown.  You should see a picture at this point.  I  recommend turning off mirrored displays if it is enabled (more about this below).

You may have to play with the underscan/overscan settings on your TV and the displays prefs for the TV if the picture is too small or larger for the TV screen.

Also, you need to set your Sound preferences to output to the HDMI connection.

And if running Lion, do not run in Lion full screen mode.

Arrangements, Menu Bar, Extended Displays, and Mirrored Displays

When you have multiple monitors (and the TV is considered just a monitor) only one monitor is designated as the "main" monitor.  That's the one with the menu bar.

If you look at the Arrangement tab in your Displays preferences you will see both monitors.  One will have a little white "menu bar" on it.  That's your main monitor.  You can drag the little menu bar to the other monitor if you wish to make it the main monitor.

You cannot make both monitors main monitors.  However there is a third party app called MenuEverywhere that allows you to do that (it has mixed reviews).

Your two monitors can be spatially arranged with the Arrangement in Displays as well.  Drag the monitors around each other to define their relationship.  Then you can drag the mouse or windows across with in that "space".  This is extended desktop mode and I recommend that to be able to use all the available screen real estate.  However there is a checkbox in the Arrangement to run in mirrored mode.  Now the two images are identical on both screens albeit the max screen size is determined by the smallest screen in the mirrored set.  You would of course have the menu bar on both screens since they are mirrors of each other.  But I think it's a waste of good screen real estate.

Are you sure the TV is set to take input from the HDMI connection port you used?  If so, and you are sure of the connections, but still no Arrangment tab in the computer's Displays preferences, there is a physical problem. 

Could be the computer so hopefully the "genius" will determine that (don't hold your breath, that's why I always quote "genius" in this context).  Could be the cable you are using (I use a mDP-to-HDMI adapter and standard HDMI cable).  Or it could be your TV (maybe it want's a specific port or you didn't set something up correctly -- check its manual).  Lot's of variables here.

Ok, good luck.