Thread: arcsin(sin(x)) ?

How do I know when $\displaystyle arcsin(sin(x))$ is

$\displaystyle x$
$\displaystyle \pi - x$ or
$\displaystyle 2\pi - x$ ?

Originally Posted by circuscircus

How do I know when $\displaystyle arcsin(sin(x))$ is

$\displaystyle x$
$\displaystyle \pi - x$ or
$\displaystyle 2\pi - x$ ?

Set $\displaystyle \alpha=\arcsin(\sin x)\implies\sin\alpha=\sin x$

The function $\displaystyle \sin^{-1} (\sin x)$ is very interesting. But it gives back $\displaystyle x$ when $\displaystyle |x| \leq \frac{\pi}{2}$.

Yea but when it's like

$\displaystyle arcsin(sin(3))$

$\displaystyle arcsin(sin(4))$

$\displaystyle arcsin(sin(5))$


I think the first one is $\displaystyle \pi-3$, second one is $\displaystyle \pi-4$ but last one is $\displaystyle \2pi-5$ (I'm going by memory from the lecture but you get what I'm saying like when it is outside that |pi/2| bound, it gets weird but I'm going to get tested on that

I don't understand why it is those values...

Have a look at this picture.
(I might find its Fourier series if I am bored. It looks nice).

I'm sorry...I don't understand graphs too well...could you explain what it means? I think I understand like have the brief idea but I can't translate the stuff I see on the chart to like pi-3, 2pi-5, etc...

Originally Posted by circuscircus

I'm sorry...I don't understand graphs too well...could you explain what it means? I think I understand like have the brief idea but I can't translate the stuff I see on the chart to like pi-3, 2pi-5, etc...

Let $\displaystyle y=\sin^{-1} (\sin x)$. It means if you got at a certain point on the x-axis say $\displaystyle x=\frac{\pi}{2}$ then the y-value is the output you get if you subsitute $\displaystyle x$ for $\displaystyle \frac{\pi}{2}$ in $\displaystyle \sin^{-1}(\sin \frac{\pi}{2} )$. The reason why I graphed it is so that you can just find all the values by just looking without calculating anything.