a question about arcsin

If I'm understanding you right, you've noticed that:

\(\displaystyle \sin \left(-\frac{\pi}{3} \right) = -\frac{\sqrt{3}}{2}\) and \(\displaystyle \sin \left(\frac{5\pi}{3} \right) = -\frac{\sqrt{3}}{2}\)

But are struggling to reconcile this information with what your calculator tells you:

\(\displaystyle \sin^{-1} \left( -\frac{\sqrt{3}}{2} \right) = -\frac{\pi}{3}\)

Assuming this is correct, the trick here lies in the fact that we define \(\displaystyle \sin^{-1}(x) = \arcsin(x)\) as the inverse function of sine. This means that for every \(x\) in the domain, there must be exactly one output. To ensure this, we restrict the domain of inverse sine to \(\displaystyle [-1, 1]\) which also necessarily restricts the range to \( [-\frac{\pi}{2}, \frac{\pi}{2}] \)

Hence, the inverse sine function maps \( -\frac{\sqrt{3}}{2} \) uniquely to \( -\frac{\pi}{3} \), rather than any of the infinitely many other \(y\) such that \( \sin(y) = -\frac{\sqrt{3}}{2} \)
 
To state it just a little differently, we define the arcsine function by "arcsin(x) is the angle [MATH]\theta[/MATH] in the interval [MATH][-\pi/2\le \theta \le\pi/2][/MATH] whose sine is x". It is not just any such angle.

We have to make such a restriction so that it will be a function; other intervals could be used, but this is what we agree to use.
 
It's not that "arcsin has no meaning", it is that "arcsin" is NOT a number so that you cannot subtract a number from it.
 
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