The reason that we use continuous sine waves today is because there are so many different people and devices that want to use radio waves at the same time. If you had some way to see them, you would find that there are literally thousands of different radio waves (in the form of sine waves) around you right now -- TV broadcasts, AM and FM radio broadcasts, police and fire radios, satellite TV transmissions, cell phone conversations, GPS signals, and so on. It is amazing how many uses there are for radio waves today (see How the Radio Spectrum Works to get an idea). Each different radio signal uses a different sine wave frequency, and that is how they are all separated.
Any radio setup has two parts:
The transmitter
The receiver
The transmitter takes some sort of message (it could be the sound of someone's voice, pictures for a TV set, data for a radio modem or whatever), encodes it onto a sine wave and transmits it with radio waves. The receiver receives the radio waves and decodes the message from the sine wave it receives. Both the transmitter and receiver use antennas to radiate and capture the radio signal.