Csc^-1(X) Function: How To Evaluate The Inverse Cosecant Function For A Given Value Of X

Csc^-1(x)

-1/|x| sqrt(x^2 -1)

The expression `csc^-1(x)` represents the inverse cosecant function. The function `csc(theta)` is defined as the reciprocal of the sine function such that `csc(theta) = 1/sin(theta)`. The inverse cosecant function `csc^-1(x)` is then defined as the angle whose cosecant is equal to `x`.

To evaluate `csc^-1(x)` for a given value of `x`, we must first find the angle `theta` such that `csc(theta) = x`. This can be done using the following steps:

1. Let `theta = csc^-1(x)`
2. Then, `csc(theta) = x`
3. Rewrite `csc(theta)` as `1/sin(theta)`.
4. Therefore, `1/sin(theta) = x`
5. Solving for `sin(theta)`, we get `sin(theta) = 1/x`
6. Using the inverse sine function, we get `theta = sin^-1(1/x)`

Therefore, `csc^-1(x) = sin^-1(1/x)`. It is important to note that the domain of `csc^-1(x)` is restricted to `x` values between `-infinity` and `-1`, and values between `1` and `infinity`. This is because the cosecant function is undefined at `0`, and the inverse cosecant function is only defined for values outside the range `[-1,1]`.

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