The definition of `cosec(phi)` is that it is equal to `1/sin(phi)` .
`cosec(2pi - theta) = 1/sin(2pi - theta)`
Now, remembering the sine curve we have that
`sin(2pi - theta) = -sin(theta)`
If `cosec(2pi-theta) = x` this implies that
`sin(theta) = -1/(cosec(2pi-theta)) = -1/x`
Therefore, in terms of `x`
`theta = sin^(-1)(-1/x)` .
Alternatively, there is the relation `cosec(phi) = sec(pi/2-phi)` so that
we have that
`cosec(2pi-theta) = sec(pi/2 - 2pi + theta) = sec(theta - 3/2pi)`
Using also the relation `sec(phi) = 1/cos(phi)` and that `cos(phi) = sin(pi/2-phi)`
`x = 1/sin(2pi - theta)`
We then continue as above, using the relation `sin(2pi-phi) = -sin(phi)`.
In terms of x
`theta = sin^(-1)(1/x)` answer
The function `theta=theta(x)`