# Express ```theta` in terms of `x` given the following expression `x = cosec(2pi-theta)`

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The definition of `cosec(phi)` is that it is equal to `1/sin(phi)` .

Therefore

`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)`