Upon entering her favorite theme park, a brilliant young princess saw a majestic castle. The castle stands $183\ feet$ tall and is $1,176\ feet$ from where the princess stood. What is the angle of elevation from the glass slippers of the princess to the top of the castle? Find the angle to the nearest hundredth of a degree.
Assume there is a perfect horizontal from the shoes to the base of the castle and that the castle is perpendicular to the ground.