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"Sonriendo" is a form of "sonreír", an intransitive verb which is often translated as "to smile". "Sonriente" is an adjective which is often translated as "smiling". Learn more about the difference between "sonriente" and "sonriendo" below.
sonriente()
An adjective is a word that describes a noun (e.g., the big dog).
1. (happy)
a. smiling (face or eyes)
La niña sonriente sostiene un ramo de flores.The smiling girl is holding a bouquet of flowers.
sonreír()
An intransitive verb is one that does not require a direct object (e.g., The man sneezed.).
2. (to favor)
a. to smile on
Parece que la vida me sonríe: al fin he encontrado un trabajo donde me valoran de verdad.It seems that life smiles on me: I have finally found a job where I'm really valued.
sonreírse
A pronominal verb always uses a reflexive pronoun. (e.g., Te ves cansado.).