We've studied the graph of in some detail in class. You might wonder, "Why do we care? Does this ever show up in real life?" The answer is most definitely "YES." Sinusoidal waves (or sine waves for short) have turned out to be essential to understanding how our world works.
One example is sound: whenever you play an instrument, or listen to your stereo, you're listening to sound waves. Guess what: those sound waves are shaped like sine waves. For example, if you know anything about playing a piano, the note A above middle C produces a wave shaped like . If you figure out the period of this function (using the theorem from class) you'll see that this wave has 440 complete cycles every second. Here's a piece of the graph:
Here's the sound itself: A [wav file]
If we double the frequency (so the sin wave completes 880 cycles per second), we get a note which sounds almost the same, and yet it's higher. If you know anything about music, we still get the note A, but now it's one octave higher. You can see the new graph and hear the sound below. (Notice how the period of this sine wave is half of the original sine wave.)
Here's the second sound: High A [wav file]