Graph and properties of inverse cosec function
Consider a function cosec y = z where y is the cosecant angle and z being the output of the cosec function, the inverse cosecant function can be written as cosec^{1}z = y. Out of the six trig ratios, cosecant of an angle is given by the ratio of hypotenuse of right triangle to its opposite side. Hence the inverse of cosecant can be given by cosec^{1}(hypotenuse / opposite) = angle θ. It can also be denoted as arccosec which means the angle made by a arc on the cosecant circle. Before moving to the examples, we will have a quick look at the inverse cosecant graph and its properties.The graph of inverse cosec function is shown below. The inverse cosec function is a multivalued function and returns explicit values in multiples of π.
Lets study some important properties of inverse cosec function.

The domain of the inverse cosec function is (∞, +∞).

The range of inverse cosecant function is (π/2, 0) to (0, π/2).

The cosecant angle of right triangle can be obtained by csc^{1}(hypotenuse / opposite).

The graph has a horizontal asymptote at x = 0 as shown.

Since the cosec is not defined at x = 0, the output of inverse function is also unavailable at that point.

It is defined for first & fourth quadrant only.
Now that we have seen the graph and key properties of inverse of cosecant, we will see few examples and explore it in a better way.
(1) Find the cosecant angle of right triangle if length of opposite side is 2 cm and that of hypotenuse is 4 cm:
We know that the cosecant angle is given by the formula θ = csc^{1}(hyp / opp) = csc^{1}(4 / 2) = csc^{1}2. But we know that the value of cosec 30^{0} is 2. Hence inverse cosecant of 2 gives 30. Thus the cosecant angle is equal to 30 degrees.
Thus we had a detailed study of inverse cosec function along with the graph and properties which will benefit a lot in solving complex trig inverse problems.
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