For what values of x is arcsin(sin x) = x?
This is for a calculus class. I'm kind of confused.
Hello, Mush89!
$\displaystyle \text{For what values of }x\text{ is }\,\arcsin(\sin x) \,=\, x\,?$
Interesting question!
Since $\displaystyle \arcsin$ always returns the principal value,
. . I would say: .$\displaystyle x \in \left[\text{-}\frac{\pi}{2},\:\frac{\pi}{2}\right]$
For other angles, say, $\displaystyle \frac{5\pi}{6}$,
. . we have: .$\displaystyle \sin\frac{5\pi}{6} = \frac{1}{2}$
. . then: .$\displaystyle \arcsin(\frac{1}{2}) \,=\,\frac{\pi}{6}$
That is: .$\displaystyle \arcsin(\sin\frac{5\pi}{6}) \:=\:\frac{\pi}{6}$