what is the principle value of invers trignometric function
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Since none of the six trigonometric functions are one-to-one, they are restricted in order to have inverse functions. Therefore the ranges of the inverse functions are proper subsets of the domains of the original functions
For example, using function in the sense of multivalued functions, just as the square root function y = √x could be defined from y2 = x, the function y = arcsin(x) is defined so that sin(y) = x. For a given real number x, with −1 ≤ x ≤ 1, there are multiple (in fact, countably infinitely many) numbers y such that sin(y) = x; for example, sin(0) = 0, but also sin(π) = 0, sin(2π) = 0, etc. When only one value is desired, the function may be restricted to its principal branch. With this restriction, for each x in the domain the expression arcsin(x) will evaluate only to a single value, called its principal value. These properties apply to all the inverse trigonometric functions.
The principal inverses are listed in the following table.
