# Will the range of an inverse trigonometric function change, if the domain of the initial function is changed?

For example, I write a function $$y=\sin x$$ and set its domain to be $$\frac{\pi}{2}.

Now, suppose I want to express $$x$$ from it, so the result is $$x=\arcsin y$$ But the domain for $$x$$ was $$\frac{\pi}{2}, while $$\arcsin y$$ outputs values between $$\left[-\frac{\pi}{2};\frac{\pi}{2}\right]$$.

Does it mean that I just changed the range of $$\arcsin y$$ to be $$\left(\frac{\pi}{2};\pi\right)$$, since $$x$$ and $$\arcsin y$$ are equal to each other in this scenario? I thought $$\arcsin$$ or any other inverse trigonometric function has a fixed range, yet here it feels I am redefining its range by setting a specific domain for the initial trigonometric function, which feels wrong, but I cannot explain why.

$$y=\sin x, \quad \frac{\pi}{2}

The function $$f:(\frac{\pi}{2},\pi) \to (1,0)$$ is one-one onto i.e. a bijection and hence it is invertable. The inverse of $$y$$ can be denoted as $$\text{myarcsin}$$.

In other words,$$\sin x=y\iff x= \text{myarcsin} \,y \text{ for all } x\in(\frac{\pi}{2},\pi) \text{ and } y\in(1,0)$$

So, clearly you do not modify the domain of $$\arcsin$$ function. You have created a new function i.e $$\text{myarcsin}$$.

The $$\arcsin$$ function is by definition inverse of $$\sin x:[-\frac{\pi}{2},\frac{\pi}{2}] \to [-1,1]$$. So, it cannot be changed.

"Now, suppose I want to express $$x$$ from it, so the result is $$x=\arcsin y$$"

It is not possible. As, $$x\in(\frac{\pi}{2},\pi)$$. But you can create a inverse function as I mention in the answer. You can see the graph.

• I think I get it, thank you! Just to clarify, what is "a bijection"?
– Tom
Commented Jan 5 at 14:57
• @Tom Bijection is term related to functions. In simple words, for every element in domain if we get a unique element in co-domain and co-domain doesnot contain any other element than these, then It is bijection or bijective function and one can only take inverse of bijection function.
– O M
Commented Jan 5 at 15:20