I once had a bible in which I had marked every instance of the words "rejoice" with a symbol reflecting the original Hebrew or Greek word which had been translated as such. There are several deeper meanings to the word, and as worshipers, I believe it is important that we are aware of the fullness of the concept, as seen in various scriptures. Linking to the original Strongs from the King James version of the Bible, I will list them here, in simple order of Strong's number:
Strong's H1523 - giyl: to spin round (under the influence of any violent emotion), i.e. usually rejoice, or (as cringing) fear:--be glad, joy, be joyful, rejoice.
Strong's H1524 - giyl: a revolution (of time, i.e. an age); also joy:--X exceedingly, gladness, X greatly, joy, rejoice(-ing), sort.
Strong's H5965 - `alac: to leap for joy, i.e. exult, wave joyously:--X peacock, rejoice, solace self.
Strong's H5970 - `alats: to jump for joy, i.e. exult:--be joyful, rejoice, triumph.
Strong's H7440 - rinnah:properly, a creaking (or shrill sound), i.e. shout (of joy or grief):--cry, gladness, joy, proclamation, rejoicing, shouting, sing(-ing), triumph (Note: this word is not found as "rejoice" but only as "rejoicing" in the King James).
Strong's H7442 - ranan: properly, to creak (or emit a stridulous sound), i.e. to shout (usually for joy):--aloud for joy, cry out, be joyful (greatly, make to) rejoice, (cause to) shout (for joy), (cause to) sing (aloud, for joy, out), triumph.
Strong's H7797 - suws: to be bright, i.e. cheerful:--be glad, X greatly, joy, make mirth, rejoice.
Strong's H7832 - sachaq: to laugh (in pleasure or detraction); by implication, to play:--deride, have in derision, laugh, make merry, mock(-er), play, rejoice, (laugh to) scorn, be in (make) sport.
Strong's H8055 - samach: probably to brighten up, i.e. (figuratively) be (causatively, make) blithe or gleesome:--cheer up, be (make) glad, (have, make) joy(-ful), be (make) merry, (cause to, make to) rejoice, X very.
Strong's H8057 - simchah: blithesomeness or glee, (religious or festival):--X exceeding(-ly), gladness, joy(-fulness), mirth, pleasure, rejoice(-ing).
Notes: an "X" in a definition refers to a rendering in the King James that results from an idiom peculiar to the original Hebrew.
Though some are clearly from similar origins, we see that there are 16 unique Hebrew words translated "Rejoice" (or rejoiceth, or rejoicing, or rejoiced) in the King James Bible. They do not all simply mean "be happy" or "put a smile on your face." Some of the words have implications of action. Much like fans at a football game, there are times when it is Biblically appropriate to be exuberant in our rejoicing. We are at times to "jump for joy" or "shout" and "laugh."
My personal favorite is the first one on the list. Here we see several examples of scripture where in rejoicing, it is even scriptural to "spin around" in our excitement of Him.