Shall we cook dinner? U use "shall" For inviting People to do something by an informal way. It's the same meaning of "would u like to cook dinner? " but less formal

You don't really need to use this word typically. As Nicole stated, it can mean "should" when beginning a question. In this case, it is a polite way of asking a question: "Shall I bring something to the dinner?" It is also used for things that are commanded by a rule: "You shall not pass." In both cases, you could kind of think of it as referring to an order, except it is only considered polite to order yourself around. You can say "should" or "may" for questions. "Should I bring something to the dinner?" means would you prefer it if I did? "May I bring something to the dinner?" means I'd really like to, would it get in the way? "May" works better if you really want to bring something. You can say "should", "will", or "had better" for commands. "You should move!" "You had better move!" "You will move now!" As rude as these statements sound, saying "You shall move!" is bizarre, because it sounds as if you are commanding the cosmic order.