cheer:  Originally, cheer meant ‘face’. It came via Anglo-Norman chere ‘face’ and late Latin cara ‘face’ from Greek kárā ‘head’. As often happens, ‘face’ was taken as a metaphor of the mental condition causing the expression on it, so ‘be of good cheer’ came to mean ‘be in a good mood’; and gradually cheer grew to be used on its own for ‘happy frame of mind, cheerfulness’. It first appears in the sense ‘shout of applause or encouragement’ at the start of the 18th century, when Daniel Defoe identifies it as a nautical usage.
c. 1200, "the face," especially as expressing emotion, from Anglo-French chere "the face," Old French chiere "face, countenance, look, expression," from Late Latin cara "face" (source also of Spanish cara), possibly from Greek kara "head," from PIE root *ker- (1) "head, horn" (see horn (n.)). From mid-13c. as "frame of mind, state of feeling, spirit; mood, humor."
By late 14c. the meaning had extended metaphorically to "mood, mental condition," as reflected in the face. This could be in a good or bad sense ("The feend ... beguiled her with treacherye, and brought her into a dreerye cheere," "Merline," c. 1500), but a positive sense (probably short for good cheer) has predominated since c. 1400. Meaning "shout of encouragement" first recorded 1720, perhaps nautical slang (compare earlier verbal sense, "to encourage by words or deeds," early 15c.). The antique English greeting what cheer (mid-15c.) was picked up by Algonquian Indians of southern New England from the Puritans and spread in Indian languages as far as Canada.
late 14c., "to cheer up, humor, console;" c. 1400 as "entertain with food or drink," from cheer (n.). Related: Cheered; cheering. Sense of "to encourage by words or deeds" is early 15c. Which had focused to "salute with shouts of applause" by late 18c. Cheer up (intransitive) first attested 1670s.