What is csc(180 degrees)?

Undefined

##csc(x) = 1/sin(x)## so

##csc(180°) = 1/sin(180°)= 1/0##. Because we're dividing by 0, it is undefined.

To understand why this strange answer makes sense, think about the unit circle: graph{x^2+y^2=1 [-2.5, 2.5, -1.25, 1.25]}

If we tried to make a right triangle on the unit circle with points at the origin, or ##(0,0)##, and the point ##(1,0)## with an angle of 180°, our third point would be ##(-1,0)##. This sad triangle would just be a line segment!

Cosecant is also used to find the ratio of hypotenuse/opposite side of a right triangle. The "opposite side" of this triangle doesn't exist, so we call this cosecant undefined.