# arcsin(sin x) = x

• Mar 8th 2011, 08:23 AM
Mush89
arcsin(sin x) = x
For what values of x is arcsin(sin x) = x?
This is for a calculus class. I'm kind of confused.
• Mar 8th 2011, 08:33 AM
Plato
Quote:

Originally Posted by Mush89
For what values of x is arcsin(sin x) = x?
This is for a calculus class. I'm kind of confused.

I suggest that you plot both these functions:
$\displaystyle f(x)=\arcsin(\sin(x))~\&~g(x)=x$.
Where do the graphs agree. And why?
• Mar 8th 2011, 11:54 AM
Soroban
Hello, Mush89!

Quote:

$\displaystyle \text{For what values of }x\text{ is }\,\arcsin(\sin x) \,=\, x\,?$

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: .$\displaystyle \arcsin(\sin\frac{5\pi}{6}) \:=\:\frac{\pi}{6}$