Optimistically / äptəˈmistiklē / adverb

Definition:  Optimistically means in a way that shows hope and confidence in the future. To be optimistic about a situation generally means hoping that it turns out for the good or in a good way.

Etymology: The term derives from the Latin optimum, which means the best. Being optimistic refers to expecting the best possible result from a particular situation.

In a Sentence

She applied for the job optimistically and waited for a placement.

John optimistically took the call; he was certain he would get the answers he was looking for. 

The doctor optimistically feels that she will be in full health soon.   

Synonyms

Hopefully, Joyfully, Amusedly, Giddily

Antonyms 

Pessimistically, Negatively, Gloomily, Ominous.