Question

What is csc(180 degrees)?

Answer 1

Undefined

Explanation:

`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.