# Domain/Range of Inverse Trig Function

• Jan 10th 2011, 07:03 PM
cusseta
Domain/Range of Inverse Trig Function
This question is difficult to put into words, but here we go...

1) If the domain of y = sin(x) is all possible x values...
2) And if the range of y = sin(x) is all possible y values...

...When looking at the inverse of this function...

3) Wouldn't the domain of x = sin(y) be all possible y values?...
4) And, similarly, the range of x = sin(y) be all possible x values?...

...Regardless of the notation being y = arcsin(x) or y = sin^-1(x)

I ask because I'm confused as to why the y values of x = sin(y) are always referred to as the range in this situation (and x values as the domain). Maybe I'm missing something huge, but it seems to me that the common notation (y = arcsin(x)) should not take precedence over the actual meaning (x = sin(y)) when deciding upon the domain and range of a function. Wow, that was difficult to express! (Worried)
• Jan 10th 2011, 07:12 PM
mr fantastic
Quote:

Originally Posted by cusseta
This question is difficult to put into words, but here we go...

1) If the domain of y = sin(x) is all possible x values...
2) And if the range of y = sin(x) is all possible y values...

...When looking at the inverse of this function...

3) Wouldn't the domain of x = sin(y) be all possible y values?...
4) And, similarly, the range of x = sin(y) be all possible x values?...

...Regardless of the notation being y = arcsin(x) or y = sin^-1(x)

I ask because I'm confused as to why the y values of x = sin(y) are always referred to as the range in this situation (and x values as the domain). Maybe I'm missing something huge, but it seems to me that the common notation (y = arcsin(x)) should not take precedence over the actual meaning (x = sin(y)) when deciding upon the domain and range of a function. Wow, that was difficult to express! (Worried)

If you want the inverse function $\displaystyle \sin^{-1}(x)$ to exist, then the domain of sin(x) must be restricted so that it is a 1-to-1 function. The conventional restriction is $\displaystyle \displaystyle -\frac{\pi}{2} \leq x \leq \frac{\pi}{2}$.