Which of the following are correct?
- Will let you know in the afternoon.
- Will let you know afternoon.
- Will let you know after noon.
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This sentence is correct. The definite article defines which afternoon you're talking about, therefore there's nothing missing.
Here the definition of which afternoon you have in mind is missing. Possible ways of correcting the sentence:
This sentence is technically correct, but the meaning is different. Afternoon is defined as the time from noon (12:00) to 18:00, whereas when you say after noon, you imply right after noon = right after 12:00. The time is limited closely to after 12:00. Similar to as when you say:
Here you say that after you finish eating your lunch, you will go on doing the promised thing. The same would apply to "after noon."
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Number two is incorrect.
Another common correct alternative is "this afternoon".
1 & 3 are acceptable.
2 is incorrect because afternoon is not the object of the verb "know". For it to be grammatically correct, the meaning would have to be "I will teach you what afternoon is like." Therefore you need a preposition to specify the relationship between afternoon and the rest of the sentence.