I have got a thermometer with a remore temperature sensor that sends
radio signals to update the base unit's display. The base unit also
has its own temperature sensor to let it display the indoors
temperature as well as the sensor's temperature.

The sender unit's antenna is a coil of wire which is about 1/4-inch
diameter and a couple of inches long. See these pics:

http://www.flickr.com/photos/7472570@N06/430806010/
http://www.flickr.com/photos/7472570@N06/430806012/

The base unit's antenna is just a short length of wire. See these
pics:

http://www.flickr.com/photos/7472570@N06/430806014/
http://www.flickr.com/photos/7472570@N06/430806016/

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TWO QUESTIONS: What is the best orientation for each of these units
so their antennas are postioned optimally? For simplicity I am
assuming no obstructions.

TWO MORE QUESTIONS: which component in the sender is the temperature
sensor? The sender's case seems sealed with no air vent!

Similarly, which component in the base unit is the sensor for the
indoors temperature? You can see the air vents in the base unit in
one picture linked above but which component is the temp sensor?