Why is "sad" not a verb? [closed]

I'm learning English grammar now for the first time, and I find it extremely confusing.

A verb is defined as

an action, an occurrence, or a state of being. Whether mental, physical, or mechanical, verbs always express activity.

If verbs are about a state of being, then why isn't "sad" a verb? Isn't it a state of being? Why is it any different from "recognized", which is a verb because it describes a cognitive state?

I don't know where your definition of 'verb' comes from, but it only defines the semantic aspect of the word.

Here's a more complete definition from Oxford:

A word used to describe an action, state, or occurrence, and forming the main part of the predicate of a sentence, such as hear, become, happen.

The boldfaced portion defines the syntactic aspect of the word.

Let me explain this with this example sentence:

You look sad.

Here, the subject is 'You' and the predicate is 'look sad'. And syntactically, the main part of the predicate is 'look' and this is a verb.

Now, 'sad' here is not a verb, because syntactically it is not the main part of the predicate. Rather, 'sad' is called a complement of the verb 'look', because syntactically it is governed by the verb and completes the meaning of the predicate.

"Sad" is a verb, but at this point it is both rare and nonstandard. The OED lists the following as one definition:

To make sorrowful; to sadden. Now rare.
_{In later use only in regional or nonstandard speech.}

It gives some old examples, like this one:

May it not sad your thoughts.
_{Antonios Reuenge, 1602}

In addition, I'm able to find some very recent examples:

I was sadded to hear of the passing of Lou.
_{Memorial page for Louis S. Josselyn, Jr, 2017}

Note that to the vast majority of native speakers (myself included), "sad" is not not a verb, and trying to use it as one sounds weird and wrong. The verb you should use is "sadden".