The students said that the teacher said to use the radian button on the calculator.  Why? So that they can correctly work the problems? What about a few words what radians are and why radians?

 

The circumference of a circle is C = 2π r, where r is the radius. The length of a quarter arc is ¼ this, or ½π r. In general, the length of the arc s is s = rθ, where the angles are in radians.  We see how natural radians are, as rθ is very simple.

 

There are 360° in a circle. Why 360? Because this is what they decided. It is not a natural unit, like radians.