A five octave range is almost nonexistent in most voices. Why aren't you HAPPY with a 3.5 octave range? That is much greater then average. The comments that say you can GAIN an octave with some technique isn't true. I have studied vocal pedagogy. You are BORN with a range, and the only thing you can do with what you are born with is discover it and cultivate it. There is no such thing as "increasing" your range per say. It is really about finding your complete range. If you have been studying for a while, you've probably already found it. If you push hard for 5 octaves, you will find yourself with some unnecessary vocal damage that you don't want.
Let me explain this to you without getting all theoretical. That might get confusing to someone who hasn't studied music in detail. An octave is pretty easy to understand on a keyboard if you can read music simplistically. Hopefully your coach has given you the basics on reading notes (a good coach will if you haven't had the background in it). Let me try to explain how octaves function. I will use middle C as a basis in the concept, since most people know where middle C is on a keyboard. An octave is an interval of 8 between notes. All octave share the same letter and shares the same pitch qualities, but one is lower than the other. For example. Try this on a piano or keyboard. Play middle C. An octave higher is the note C above middle C on the keyboard, and an octave lower is the note C below middle C. And this is the same rule for notes D, E,F,G,A and all the sharps and flats between. If the notes don't share the exact same letter, its not an octave. There are frequencies involved that I am not going to explain because its too complicated, but you can definitely hear it. Just get the basic concept on the piano. So say a person has a 2 and a half octave range. Pretend they're lowest note is F below middle C. The next octave they sing is F above middle C (that's one octave), then the next octave they can sing is F above that F as mentioned before (that's two octaves). But then all of the sudden, their voice can barely hit that D above that F last mentioned. SOOOO that makes about 2 octaves and a half.
More importantly, as a person with a pretty wide range myself, let me tell you range isn't as important music wise as you think. What is more important is the quality of your voice, and the musicality you give. The only real advantage you have over a brilliant singer with an average range is that you "MIGHT" possibly be able to sing more roles. But it is danger to that for most, because constantly singing out of your vocal fach (that's another subject. Even those with wide ranges has their HOME place in the voice) can damage you in the long run. Only a few wise singers who have studied for a while and TRULY know their voice can get away with that (like Placido Domingo). So develop what you have. Be proud of it! Most people will be in awe of the range God has given to you. You really don't need a 5 octave range. There aren't ANY songs, pieces, arias, anything that causes for such a range. Perhaps as you continue to study, you might find some more notes in your range as you grow as a singer, but it is HIGHLY unlikely you will get up to a 5 range octave. I think when you listen to how wickedly WIDE it is between one octave to another (so imagine it between 5), and once you understand how octaves function, you will totally understand what I am talking about. Just try it on a piano. You will see how insanely high and low it is between 5 octaves in the human voice. Be grateful for your range and sing to the heavens!