Domain: -1 ≤ x ≤ 1

Mary Herrera

Answered 2021-12-27
Author has **4127** answers

user_27qwe

Answered 2021-12-30
Author has **9558** answers

The original sine function defined for any real argument does not have an inverse function because it does not establish a one-to-one correspondence between its domain and a range.

To be able to define an inverse function, we have to reduce the original definition of a sine function to an interval where this correspondence does take place. Any interval where sine is monotonic and takes all values in its range would fit this purpose.

For a function \(y=\sin x\) an interval of monotonic behavior is usually chosen as \([-\frac{\pi}{2},\frac{\pi}{2}],\) where the function is monotonously increasing from -1 to 1.

This variant of a sine function, reduced to an interval where it is monotonous and fills an entire range, has an inverse function called \(y=\arcsin (x)\)

It has range \([-\frac{\pi}{2},\frac{\pi}{2}]\) and domain from -1 to 1

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