Question

How do you find the exact value of `csc(theta) = 1/4`?

Answer 1

The equation has no solutions.

Explanation:

You probably mean to find the solution of the equation. And the solution doesn't exist, because `csc(x)=1/sin(x)`, and so the equation becomes

`1/sin(x)=1/4`, from which we would derive `sin(x)=4`, which is obviously impossible.

Indeed, since `sin(x) \in [-1,1]` for any `x`, then `1/sin(x)` ranges in the intervals `(-infty,-1] \cup [1,\infty)`, and so it can never be equal to `1/4`.