In The Odyssey, what are some examples that prove that Odysseus is arrogant?I already have the part of when Odysseus yells his name to the Cyclops. 

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lsumner eNotes educator| Certified Educator

The first time Odysseus showed his arrogance is when he won the battle at Troy. He yells out to the gods that he doesn't need them. Also, he does not give a sacrifice to the gods for helping the Greeks win the victory. This is arrogant behavior. This is arrogant because it shows that he doesn't need the help of the gods. In return for his arrogance, the gods create a difficult journey home for Odysseus. 

Also, Odysseus shows his arrogance again when he blames the situation solely on his men. He calls them mutinous fools. In book nine, on page 213, line 50, Odysseus talks of his crew cruelly:

"Then I urged them to cut and run, set sail
But would they listen? Not those mutinous fools.
Too much wine to swill, too many sheep to slaughter."
He blames the situation completely on his crew, taking no responsibility like a mature person, and as the crew's captain. 
In this passage, Odysseus is arrogant. He blames everything on his crew. He is self centered, self righteous and all important. His behavior is obnoxious. He does not appreciate all that his men have been through while trying to get home.  
Odysseus is arrogant when he listens to the Sirens singing. He does not put wax in his own ears. He takes a chance and could have caused his crew great harm. 
Likewise, Odysseus yells out his name to the Cyclops. As Odysseus and his men are safely escaping the Cyclops, Odysseus yells out the following:
If any man on the face of the earth should ask you
Who blinded you, shamed u so,- say Odysseus
Raider of cities, HE gouged your eyes."
Now, the Cyclops knows who blinded him. He prays to his father Poseidon for help. Odysseus loses more men because of his arrogance. 
In the end, Odysseus is humbly on raft about to drown and he cries out to the gods for help. The gods save him. 
jeremydodson | Student

He yells back at the cyclops.

When he decides he can handle listening to the Sirens, but the rest of the crew need to be "protected" from it.