a) implement additive, AM and FM synthesis using the sin wave code as a starting point.
b) using additive synthesis, create the most harmonious, beautiful sound that you can.  think about using multiple sin osicllators, with various levels of volume.  Be very, very, very careful not to clip your sounds (ie, scale the volumes so that they add up to less then 1.0).   Use many oscillators.
c) using additive synthesis, create the most evil, terrible, cacophonous and dissonant sound.  Again, take care about volume levels.
d) try to see what the limit of your computer is by adding as many sin oscillators as you can together. Â how many can you put into the app with out it skipping. Â take care about the volume of each osciallator.
e) using any synthesis technique, attempt to recreate these tones:Â
f) make music ! try to transcribe a song (turning the notes into frequencies) and play it via code. And / or, try to create a sequencer that allow you to compose music by controlling some parameters of synthesis over time.  then, create some music with this tool.
have fun!